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THE  UNIVERSITY  OF  CHICAGO 
MATHEMATICAL  SERIES 

ELIAKIM  HASTINGS  MOORE 
GENERAL  EDITOR 


SCHOOL  OF  EDUCATION 
TEXTS  AND  MANUALS 

GEORGE  WILLIAM  MYERS 
EDITOR 


FIRST-YEAR   MATHEMATICS 
FOR    SECONDARY   SCHOOLS 


THE  UNIVERSITY  OF  CHICAGO  PRESh 
CHICAGO.  ILLINOIS 

Bgente 
THE  BAKER  &  TAYLOR  COMPANY 

NEW    TOBK 

CAMBRIDGE  UNIVERSITY  PRESS 

LONDON    AND    EDINBDRaH 


First -Year  Mathematics 

For  Secondary  Schools 


By 

GEORGE  WILLIAM  M.YERS 

Professor  of  the  Teaching  of  Mathematics  and  Astronomy,  College  ot 
Education  of  the  University  of  Chicago 

and 

WILLIAM  R.  WICKES  HARRIS  F.  MacNEISH 

ERNST  R.  BRESLICH  ERNEST  A.  WREIDT 

ERNEST  L.  CALDWELL         ARNOLD  DRESDEN 

Instructors  in  Mathematics  in  the  University  High  School 
of  the  University  of  Chicago 

2/  2.8  ^ 


School  of  Education   Manuals 
Secondary  Texts 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


Copyright  1906  By 

Geo.  W.  Myebs 
COPYEIGHT  1909  By 

The  Dniveesity  of  Chicago 


First  edition  privately  printed  October,  1906 

Second  impression  published  April,  1907 

Second  edition  August,  1907 

Third  edition  September,  1509 

Second  impression  December,  1910 


Composed  and  Printed  B7 

The  Dnivenity  of  Chicago  Preu 

Chicago,  Illinoii,  U.S.I. 


TABLE  OF  CONTENTS 

CHAPTER  PAGE 

Preface ix 

I.    General  Uses  of  the  Equation    ....        i 

II.    Uses  of  the  Equation  with  Perimeters  and 

Areas 14 

III.  The  Equation  Applied  to  Angles      ...       30 

IV.  Positive  and  Negative  Numbers    ....       62 
V.    Beam  Problems  in  One  and  Two  Unknowns      89 

VI.    Problems  in  Proportion  and  Similarity.  107 

VII.    Problems    on    Parallel   Lines.     Geometric 

Constructions 137 

VIII.    The   Fundamental  Operations   Applied   to 

Integral  Algebraic  Expressions  .     .  .     153 

IX.    Practice  in  Algebraic  Language.     General 

Arithmetic 181 

X.    The  Simple  Equation  in  One  Unknown.  210 

XI.  Linear  Equations  Containing  Two  or  More 
Unknown  Numbers.  Graphic  Solution,  of 
Equations  and  Problems 234 

XII.    Fractions 262 

XIII.    Factoring.    Quadratics.    Radicals    .     .     .     275 

XIV.   Polygons.    Congruent  Triangles.    Radicals    323 

Summaries  at  the  ends  of  the  first  thirteen  chapters,  see 
pp.  13,  28-29,  59-61,  88,  106,  136,  151-52,  180, 
207-9.  232-33.  260-61,  273-74,  321-22. 

vii 


PREFACE 

This  volume  aims  to  present  in  teachable  form  an  inter- 
weaving of  the  more  concrete  and  the  easier  portions  of  the 
first  courses  in  both  algebra  and  geometry.  It  is  intended 
as  a  text  for  students  of  the  first  year  of  secondary  schools. 
The  type  of  work  begun  here  will  be  continued  by  a  second- 
year  text,  to  appear  soon  and  to  include  not  less  than  the  rest 
of  the  subject-matters  that  make  up  the  customary  two  years 
of  required  secondary  work  in  algebra  and  plane  geometry. 

The  present  book  places  chief  emphasis  upon  algebra,  but 
weaves  in  a  considerable  body  of  related  fundamental  notions 
and  principles  of  geometry,  even  advancing  somewhat  in  one 
direction  or  another  upon  the  most  concrete,  graphic,  and 
practical  aspects  of  elementary  geometry.  Geometrical  treat- 
ments are  at  first  intuitive,  inductive,  and  experimental,  fol- 
lowed in  a  few  instances  by  the  quasi-e!^erimental  methods 
of  superposition. 

A  like  informaUty  as  to  method  characterizes  the  first  half 
of  the  algebra.  The  algebraic  treatments  feature  distinctly: 
(i)  the  liberal  use  of  inductions  from  arithmetic;  (2)  the  early 
and  persistent  use  of  the  equation,  both  as  an  instrument  for 
problem-solving,  and  as  a  means  of  suggesting  and  of  treating 
topics;  (3)  the  early,  varied,  and  systematic  use  of  pictured 
and  graphic  modes  of  rendering  algebraic  truths  vivid  and 
appealing  to  beginners. 

The  algebra  of  the  last  half  of  the  book  is  more  formally 
deductive  in  character.  Even  as  early  as  p.  26  the  four  pro- 
cess-axioms are  given  as  simple  statements  of  the  four  ways 
in  which  equations  have  been  changed  in  solving  arithmetical 
and  mensurational  problems.     They  are  stated  as  laws  for 


X  Preface 

using  the  equation,  and  are  given  only  as  much  generality  as 
the  antecedent  work  justifies.  Transformations  of  equations 
are  then  made  on  the  basis  of  these  laws  and  of  others  brought 
out  from  time  to  time  as  the  work  proceeds,  until  p.  216  is 
reached,  where  a  fifth  general  axiom  is  added,  and  the  five 
are  here  given  final  statement.  These  five  axioms  are  hence- 
forth made  the  reasons  for  all  changes  in  equations.  The 
root-axiom  is  added  later  when  an  occasion  to  use  it  arises. 
Thus  the  transition  from  the  informal  procedure  of  the  earlier 
part  of  the  algebra  to  the  formal  procedure  of  the  later  part 
is  gradual,  while  throughout  the  latter  half  the  reasoning 
becomes  more  and  more  highly  deductive.  This  is  in  accord 
with  what  both  classroom  practice  and  a-priori  reasoning 
show  to  be  the  natural  order  for  secondary  mathematical 
education. 

The  subject-matter  of  the  two  years  of  work  of  this  volume 
and  its  companion,  Second-Year  Mathematics,  fully  meets 
collegiate  entrance  requirements  of  one  unit  of  algebra  and 
oie  unit  of  plane  geometry.  There  is  gain  in  keeping  the 
twofold  point  of  view  continually  before  the  learner,  habituat- 
ing him  to  face  his  problems  as  mathematical  problems  rather 
than  as  algebra  problems,  or  geometry  problems.  This  re- 
quires him  to  keep  both  kinds  of  subject-matter  longer  before 
him,  and  brings  to  his  help  continually  the  reflex  light  that 
each  subject  throws  upon  the  other.  Thus  the  result  must  be 
a  superior  quality  of  collegiate  preparation.  The  college 
teacher  of  pupils  prepared  on  this  plan,  no  less  than  the  high- 
school  teacher,  will  be  gratified  by  the  results  the  work  of  such 
students  will  show. 

Colleges  do  not  usually  require,  and  they  should  not  re- 
quire, that  the  desired  units  of  credit  be  made  by  the  study 
of  algebra  and  geometry  as  separate  and  distinct  domains  of 
mathematical  truth.  Much  is  lost  educationally  by  this  plan, 
and  the  authors  are  confident  that  college  teachers  will  be 


Preface  xi 

the  first,  the  most  numerous,  and  the  wannest  friends  of  the 
plan  of  these  two  books. 

That  such  material  as  this  book  contains  furnishes  a 
better  fitting  for  the  mathematical  needs  of  daily  life  than  a 
year  of  formal  algebra  will  hardly  be  denied.  Accordingly  it 
has  no  apologies  to  make  for  its  appearance.  It  works  an 
almost  virgin  soil  in  a  field  in  which  it  elbows  no  competitors. 

On  the  other  hand,  this  book  should  not  be  looked  upon 
in  any  sense  as  a  protest  against  algebra  as  algebra,  or  geom- 
etry as  geometry.  Some  teachers  are  succeeding  measurably 
in  making  high-school  algebra  grip  the  motives  of  high-school 
boys  and  girls,  and  this  is  the  main  thing  in  mathematical 
teaching.  For  most  teachers,  however,  it  is  much  easier  to 
motivate  a  school  subject  in  a  broad  than  in  a  narrow  field. 
This  is  particularly  true  when  in  the  narrow  field  the  ideas 
are  highly  sophisticated  and  abstract,  as  is  the  case  with 
school  algebra.  Greater  freedom  and  more  informality  in 
getting  at  the  beginner's  field  of  ideas  and  fund  of  experiences 
are  desired  by  many  of  our  best  mathematical  teachers,  and 
their  number  is  increasing. 

This  text  has  been  thoroughly  seasoned  through  four  years 
of  classroom  use  in  the  University  High  School  and  purged  by 
the  ordeals  of  severe  criticism.  Six  high-school  teachers,  with 
experience  in  high-school  work  ranging  from  five  to  twenty- 
five  years,  have  participated  in  its  authorship  and  in  the 
practical  tests  of  the  classroom.  The  present  form  of  the 
book  is  a  thoroughgoing  revision  in  the  light  of  all  this  use 
and  criticism.  The  conditions  under  which  the  material  has 
been  tried  in  every  way  are  precisely  the  conditions  of  the 
public  high  school. 

The  marked  improvement  that  has  been  wrought  in  the 
mathematical  attitude  and  tone  of  the  early  classes  of  the 
University  High  School  mainly  through  the  agency  of  this  text 
justifies  strong  claims  for  its  suitability  and  worth  for  first- 


xii  Preface 

year  classes.  Finality  is,  however,  not  claimed  for  it.  It  is 
claimed  that  this  book  will  work  now,  and  will  work  easily 
and  effectively,  under  average  conditions  in  the  public  higfh 
schools  of  the  nation.  Actual  experience  under  such  conditions 
proves  beyond  the  power  of  a-priori  argument  to  disturb,  that 
without  loss  of  mathematical  sohdity  this  book  contributes  in  a 
superior  degree  to  the  much-neglected  virtues  of  mathematical 
interest,  earnestness,  and  spirit,  and  to  an  early  and  genuine 
faith  in  the  worth  of  mathematical  study. 

The  second-year  text  will  make  geometry  the  central  theme 
and  the  algebra  will  take  second  rank.  Algebraic  matters 
will  be  related  to  kindred  parts  of  geometry,  and  will,  when 
needful,  be  given  parallel  or  independent  topical  treatments. 

The  methodology  of  last  treatments  of  this  year  will  be 
increasingly  deductive  and  logical,  though  even  here  austere 
ideals  are  not  attempted.  The  .secondary  school  can  do  prof- 
itably little  more  than  rough  out  the  bolder  outlines  of  the 
sciences  of  algebra  and  geometry.  The  hold  on  the  algebra 
learned  the  first  year  will  be  maintained,  and  the  customary 
topics  well  along  into  simultaneous  quadratics  will  be  treated 
with  customary  fulness.  Plane  geometry  will  be  completed, 
with  careful  regard  to  a  smaller  number  than  is  customary  of 
very  well-worked-out  propositions. 

In  conclusion  the  authors  desire  to  render  full  acknowl- 
edgement to  Dr.  William  B.  Owen,  dean  of  the  University  High 
School,  for  his  interest  and  very  substantial  aid  in  furthering 
the  practical  side  of  this  educational  experiment.  Without 
his  assistance  and  continued  interest,  the  task  would  have 
been  difficult,  if  not  impossible.  He  was  among  the  first  to 
recognize  the  importance  of  the  function  of  the  University 
High  School  as  an  experimental  laboratory  for  secondary  edu- 
cational problems.  Any  success  that  may  come  to  this  book 
is  due  largely  to  him. 

The  Authors 


CHAPTER  I 


M££AMtmtat4& 


GENERAL  USJio  OF  THE  EQI/aTION 

2-/2  S^ 

Problems  and  Exercises  on  the  Balance 

I.  Problems  in  weighing  are  readily  solved  by  the  aid  of 
the  equation. 

I.  A  bag  of  grain  of  unknown  weight,  w  lb.,  together  with 
an  8-lb.  weight  just  balances  an  i8-lb. 
weight.     How  much  does  the  bag  of 
grain  weigh  ? 

The  problem  may  be  stated  in  an  equa- 
tion, thus: 

•u;  +  8  =  i8,  find  w. 
Suppose  8  lb.  taken  from  each  pan,  giving 

w  =  io. 
The  bag  of  grain  weighs  lo  pounds. 


Fig. 


J„,„„„f„,,f/M//Mf 


2.  Two  equal,  but  unknown,  weights  together  with  a  i-lb, 
weight  just  balance  a  i6-lb.  and  a  i-lb. 
weight  together.  How  heavy  is  each 
unknown  weight  ? 

In  the  form  of  an  equation,  the  problem 
is  stated: 

2/»  +  i  =  i7;  find  p. 

Suppose  I  lb.  to  be  taken  from  each  pan 
giving 

2/)  =  l6. 

Then 

/,=8. 


Fig. 


3.  A  man  has  only  2-lb.,  4-lb.,  and  12-lb.  weights.  He 
finds  that  three  bags  of  shot,  of  equal,  but  unknown,  weights 
together  with  two  2-lb.  weights  on  the  left  scale-pan  and  a 


First-Year  Mat  hematics 


/r,,,„/,/,fj„,t/,A. 


It„ll,n^t,„n,„nn„,„l,nl,illlll 


i2-lb.  and  a  4-Ib.  weight  on  the  ijght  scale-pan,  just  bal- 
ance. Find  the  weight  of  each  bag  of 
shot. 

In  the  language  of  the  equation,  we  may 
write 

Taking  4  lb.  from  each  pan,  we  have 

3X  =  I2. 

Fig.  3. — An  equation      Then 
is  an  expression  of  bal-  »=4. 

ance  of  values.  Each  bag  of  shot  weighs  4  pounds. 

4.  A  pail  of  berries  of  unknown  weight,  x  lb.,  pressing 
downward,  and  a  spring  balance  pulling  upward  and  showing 
4  lb.  on  the  left  side,  and  a  weight  of 
12  lb,  on  the  right  side,  just  balance. 
How  heavy  is  the  pail  of  berries  ? 

In  the  language  of  the  equation, 
»— 4  =  12 
Adding  4  lb.  to  both  sides, 
x  =  i6. 
The  pail  of  berries  weighs  16  pounds. 

5.  If  there  were  three  pails  of  berries  of  equal,  but 
unknown,  weights  and  a  spring  balance  pulling  upward  and 
showing  3  lb.  on  the  left  side,  just  holding  42  lb.  in  balance 
on  the  right,  find  the  weight  of  each  pail  of  berries. 

Using  the  equation, 
Adding  3  lb.  to  both  sides, 
Then 

Each  pail  of  berries  weighs  15  pounds. 

6.  If  a  weight  of  Zx  and  an  upward  pull  of  10  lb.  just 
balance  70  lb.  on  the  right,  find  the  weight  x. 


Fig.  4 


3«-3=42. 
3«=4S- 


General  Uses  o}  the  Equation  3 

Using  the  equation, 

Sac— 10  =  70. 
Adding  10  lb.  to  both  sides, 

8»=8o. 
Then 

»  =  io. 
The  weight,  x,  equals  10  pounds. 

2.  Any  equation,  such  as  5:x;+8  =  48,  3ic— 5  =  10,  etc.,  may 
be  regarded  as  an  expression  of  balance  between  the  numbers 
on  the  two  sides  of  the  sign  (=)  of  equahty. 

Exercise  I 
Find  the  value  of  x  in  the  following  equations : 


I. 

af+  3=  5 

8. 

4^-  5  =  23 

IS- 

8^-  7=  57 

2. 

2X—    1=    9 

9- 

SX+  7  =  28 

16. 

8^-  7=  53 

3- 

2X+    6=16 

10. 

^x-  6  =  24 

17- 

9a;+i2=  93 

4- 

ZX-\-  7  =  19 

II. 

4J(;+io=22 

18. 

gx—  8=   46 

5- 

T,X-    4=11 

12. 

4^-  5=  5 

19. 

9^;+  8=116 

6. 

3x4-  7=28 

13- 

5rc+i7  =  62 

20. 

9:jc-i8=   54 

7- 

2:x;— 11  =  21 

14. 

6x~  9  =  27 

21. 

9^-  7=   74. 

Problems  Using  Algebraic  Language 

1.  One-half  of  John's  money  is  9  cents.     How  much  money 
has  John  ? 

Let  X  stand  for  the  number  of  cents  in  John's  money.     Then,  \  oi  x 
=  9,  and  ac  =  2X9  =  18.     John  has  18  cents. 

"i  of  x"  is  written  \x,  or  -  .     The  solution  would  then  be: 
2  ■ 

\x  =  g, 
x  =  iS. 

2.  One-fourth  of  the  number  of  acres  in  a  field  is  5  acres. 
How  many  acres  are  there  in  the  field  ? 

X 

Hence 


4  First-Year  Mathematics 

3.  One-sixth  of  a  certain  number  is  8.     What  is  the  num- 
ber ?    Solve  the  problem  by  means  of  equations. 

Exercise  II 

Give  orally  the  number  for  which  the  letter  stands  in  the 
following  equations: 


I. 

a 

2. 

c 
8=7 

3- 

i=6 

7 

4- 

^=8 
9 

5- 

r 

6. 

7- 

X 

6  =  7 

8. 

a; 

6=9 

9. 

^=8 

10. 

z 

5 

24. 

25- 

9ic=io8 

^=7 

I 

-m=< 
7       ^ 

II. 

12. 

3^=18 

26. 

13- 
14. 

7x=49 
8x=32 

27. 

^s=6 
12 

15- 

6x=48 

28. 

12^  =  132 

16. 

5^=45 

29. 

IIX=I2I 

17- 

9^  =  72 

30- 

63:x;=9 

18. 

85  =  72 

31- 

72;y=8 

19. 

7^=63 

32. 

56^=7 

20. 

9); =63 

33- 

48r=8 

21. 

122  =  84 

34- 

35^  =  7 

22. 

ii(/=66 

35- 

6o<i=5o 

23- 

i2r  =  72 

36. 

1 .2»  =  i.44. 

37.  Helen  had  8  cents  and  her  father  gave  her  19  cents  more. 
Htfw  many  cents  had  she  then  ? 

38.  A  boy  had  15  cents  and  was  given  c  cents  more.  How 
many  cents  had  he  then  ? 

3.  The  sum  of  15  and  c  is  written  thus:  15 +c,  or  c-f-15. 
The  first  form  is  read  '' fifteen  plus  c"  and  the  second  form, 
"c  plus  fifteen." 


General  Uses  0}  the  Eqtcation  5 

The  difference  of  15  subtracted  from  c  is  written  c  — 15, 
and  is  read  "c  minus  fifteen." 

1.  A  man  had  22  acres  and  sold  8  acres.  How  many 
acres  had  he  left  ? 

2.  A  man  had  22  acres  and  sold  a  acres.  How  many 
acres  had  he  left  ? 

3.  A  man  had  a  acres  and  sold  b  acres.  How  many  acres 
had  he  left  ? 

4.  A  man  had  5a  acres  and  sold  2a  acres.  How  many 
acres  had  he  left  ? 

Problems  to  be  Solved  by  Arithmetic  or  Algebra 

4.  Many  problems  may  be  solved  either  by  arithmetic,  01 
by  the  use  of  the  equation.  When  the  solution  of  a  problem 
is  made  by  the  use  of  the  equation,  it  is  commonly  called  an 
algebraic  solution. 

I.  Divide  a  pole  20  ft.  long  into  two  parts  so  that  one  part 
shall  be  4  times  as  long  as  the  other. 

Arithmetic  Solution 

The  shorter  part  is  a,  certain  length. 

The  longer  part  is  four  times  this  length 

The  whole  pole  is  then  five  times  as  long  as  the  shorter  pari. 

The  pole  is  20  ft.  long. 

The  shorter  part  is  ^  of  20  ft.,  or  4  ft. 

The  longer  part  is  4*4  ft.,  or  16  ft. 

Hence,  the  parts  are  4  ft.  and  16  ft.  long. 

Algebraic  Solution 

Let  X  be  the  number  of  feet  in  the  shorter  part,  % 

then  4X  is  the  number  of  feet  in  the  longer  part, 

and  X  +  4X,  or  ^x  —  20, 

x=  4, 

455  =  16. 

Hence,  the  parts  are  4  ft.  and  16  ft.  long. 


leet  in 

f 


2,  A  farmer  wishes  to  inclose  a  rectangular  pen  with  80  ft 


6  First-Year  Mathematics 

of  wire  fencing.     He  wishes  it  to  be  three  times  as  long  as  it 
is  wide.    How  long  shall  he  make  each  side  ? 

Algebraic  Method 

Let  X  be  the  niimber  of  feet  in  the  smaller  side, 
then  ^x  is  the  number  of  feet  in  the  longer  side, 
and  x+ye,  or  4:1;  is  the  number  of  feet  half-way  round  the  pen. 

«  =  10, 

Hence,  the  sides  are  10  feet  and  30  feet  long. 

3.  A  boy  sold  a  certain  number  of  newspapers  on  Monday, 
twice  as  many  on  Tuesday,  10  more  on  Wednesday  than  on 
Monday,  and  24  on  Thursday.  He  sold  94  in  the  four  days. 
How  many  did  he  sell  on  each  day  ? 

4.  A  man  divides  up  his  160-acre  farm  as  follows:  He 
takes  a  certain  number  of  acres  for  lots,  4  times  as  much  for 
pasture,  4  times  as  much  for  com  as  for  pasture,  ^  as  much 
for  wheat  as  for  corn,  and  15  acres  for  meadow.  How  many 
acres  does  he  assign  to  each  purpose  ? 

5.  A  book  dealer  has  in  stock  twice  as  many  Readers  as 
Arithmetics,  four  times  as  many  Readers  as  Histories.  In  aU 
he  has  70  Readers,  Arithmetics,  and  Histories.  How  many 
of  each  has  he  ?  M  ' 

6.  A  may-pole  22  ft.  high  breaks  into  two  pieces  so  that 
the  top  piece,  hanging  beside  the  lower  piece,  lacks  6  ft.  of 
reaching  the  ground.     How  long  is  e^h  piece  ? 

7.  A  pony,  a  saddle,  and  a  bridle  together  cost  $120.  The 
bridle  costs  ^  as  much  as  the  saddJ^^pd  the  pony  costs  $12 
less  than  I2»times  as  much  as  the  s^He.  What  was  the  cost 
of  each  ?  4 

8.  A  bicyclist  rode  a  certain  number  of  miles  on  Monday, 
f  as  many  miles  on  Tuesday,  f  as  many  on  Wednesday,  f  as 


General  Uses  of  the  Equation  7 

many  on  Thursday,  as  many  on  Friday  as  on  Monday,  and 
20  miles  on  Saturday.  On  the  six  days  he  rode  152  miles. 
How  many  miles  did  he  ride  each  day  ? 

9.  James  has  3  times  as  many  cents  as  Charles,  and  4 
times  as  many  as  William.  All  together  they  have  57  cents. 
How  many  cents  has  each  ? 

10.  The  area  of  a  triangular  piece  of  ground  is  315  sq.  rd. 
One  side  is  30  rods.  How  long  is  a  fence  at  right  angles  to 
this  side  from  the  opposite  comer  ?     (Use  principle  below.) 

Principle. — The  area  0}  a  triangle  is  equal  to  ^  the  product 
of  its  base  and  altitude. 

11.  If  this  fence  divides  the  side  (30  rods)  so  that  one  part 
is  twice  as  long  as  the  other,  what  are  the  areas  of  the  two 
lots? 

12.  If  the  fence  divides  the  side  (30  rods)  so  that  one  part 
is  five  times  the  other,  what  are  the  areas  of  the  two  lots  ? 

13.  A  local  train  goes  at  the  rate  of  30  miles  an  hour.  An 
express  starts  two  hours  fater  and  goes  at  the  rate  of  50  miles 
an  hour.  In  how  many  hours  and  how  far  from  the  starting- 
point  will  the  second  train  overtake  the  first  ? 

14.  If  75  dollars  are  to  be  divided  between  two  persons 
so  that  one  shall  have  27  dollars  more  than  the  other,  how 
much  must  each  one  have  ? 

15.  Three  men.  A,  B,  and  C,  wish  to  divide  1,584  shares 
of  stock  among  themselves  so  that  A  shall  have  25  more  than 
B,  and  C  shall  have  50  more  than  B.  How  many  shares 
must  each  receive  ? 

16.  At  an  election  in  the  Fxeshman  class  of  the  University 
High  School  84  votes  were  cast.  Candidates  A  and  B  each 
received  a  certain  number  of  votes;  candidates  C  and  D  each 
received  three  times  as  many  as  candidate  A;   and  candidate 


8  First-Year  Mathematics 

E  received  four  times  as  many  as  D,  plus  4.    How  many  votes 
did  each  candidate  receive  ? 

17.  A  regulation  foot-ball  field  is  56§  yards  longer  than  it 
is  wide  and  the  sum  of  its  length  and  width  is  163  J  yards. 
]^d  its  dimensions. 

18.  The  manager  of  a  high-school  foot-ball  team  paid  the 
manager  of  a  visiting  team  $63  for  expenses.  If  the  payment 
was  made  in  $2  and  $5  bills,  the  same  number  of  each,  how 
many  bills  of  each  kind  were  paid  ? 

19.  A  line  40  ft.  long  is  divided  so  that  the  longer  part  is 
12  ft.  more  than  3  times  as  long  as  the  shorter  part.  Find 
the  length  of  each  part.  Draw  a  line  4  inches  long  to  represent 
the  40  ft.  Une,  and  mark  on  it  the  lengths  of  the  two  parts. 

20.  Two  men,  A  and  B,  wish  to  divide  $5,247  betw^een 
them  so  that  B  shall  receive  $324  more  than  twice  as  much  as 
A.    How  much  should  each  receive  ? 

21.  The  length  of  a  rectangle  is  26  yd.  greater  than  the 
width.  If  the  perimeter  (distance  around)  is  432  yd.,  find 
the  dimensions. 

22.  Find  two  consecutive  numbers  whose  sum  is  203. 
(Suggestion:   Let  x  be  one  number,  and  :x;-|-i  the  other.) 

23.  Find  three  consecutive  numbers  whose  sum  is  474. 

24.  Find  two  consecutive  odd  numbers  whose  sum  is  156. 
(Suggestion-  Let  x  be  one  number  and  x-\-2  the  other.) 

25.  Find  two  consecutive  even  numbers  whose  sum  is  378. 

26.  Find  three  consecutive  even  numbers  whose  sum  is 
372. 

27.  A  college  student  spent  during  his  Freshman  year  S620 
for  tuition,  room  and  board,  and  books.  For  tuition  he  spent 
four  times  as  much  as  he  did  for  books,  plus  $20,     For  room 


General  Uses  of  the  Equation  g 

and  board  together  he  spent  ten  times  as  much  as  for  books. 
How  much  did  he  spend  for  each  item  ? 

28.  The  tuition  at  the  University  High  School  is  $150 
a  year.  If  this  tuition  is  paid  in  twenty-dollar  bills,  and  if 
the  same  number  of  five-dollar  bills  is  received  in  change,  how 
many  twenty-dollar  bills  are  paid  ? 

29.  Divide  341  into  three  parts  so  that  the  second  part 
shall  be  five  times  the  first  part,  and  the  third  part  shall  be 
two  times  the  first  part  plus  5. 

30.  A  farmer  wishes  to  lay  out  a  field  in  4.he  shape  of  a 
rectangle  whose  width  is  f  of  its  length,  and  whose  perimeter 
is  320.     How  long  and  how  wide  must  he  make  the  field  ? 

Exercise  HI 
The  following  exercises  test  and  review  the  addition  and 
multiplication  facts  of  arithmetic,  and  give  exercise  in  alge- 
braic equations.     Find  the  value  of  the  letter,  x,  doing  all  you 
can  orally: 

1.  2:x;-f-x=9  16.  8x— 8=64 

2.  ;^x—x=8  17.  8x— 2:x;=42 

3.  2x-|-io=22  18.  8:»-fai;=8i 

4.  2:x;— 7  =  13  19.  8;x;+2:x:=7o 

5.  75;  — 8=48  20.    8x  —  2X  =  'J2 

6.  7ic+i=64  21.  ^x+2X+x=i5 

7.  7j;-|-6=62  22.  4:x;-f:)(;— 25£;=2i 

8.  7:^—9=40  23.  6x+'jx—^x=So 

9.  'jx+2x=2'j  24.  8:>i;-f-4X— 7:x;=6o 

10.  'jx—x=;^6  25.  5^-f-9X-f-2:x;=64 

11.  75c— 25£;=6o  26.  9:x;—4:)£;— 2:^=36 

12.  4:x;-|-5:v=8i  27.  iix—2x+;^x=io8 

13.  8:x; -1-2=66  28.  5X+6x+'jx=^6 

14.  8j(;— 4=62  29.  'jx-\-2x—S=46 

15.  8;x; -1-9=81  30.  7:x;  — 2;>;+6=4i 


lo  First-Year  Mathematics 

31.  I2X— 4:»+x=9o  41.  ^x+'jx+i5x—2X-{-^  =  'j4 

32.  2X—X  +  ^X  =  20  42.    2X-\-JX—^X  —  6=24. 

33.  651; +^-331; =18  43.  {x=6 

34.  yx-x+sx^Tj  44.  i^=3 

35.  i2X+8x-sx=4S  45-  §^=8 

36.  i8jc— 9X+2:x;=66  46.  |3£;=2o 

37.  igx—Sjc— 3^=56  47.  x-^^x=6 
.     38.  205(;— 85£;+:v=26  48.  x—^x  =  'j 

39.  1451; +4^— 9^=63         49.  fx+i-'c— i^=i8 

40.  i2ap+9x— 3J(;=63         50.  6af— §:»— iae+7  =  i29. 

Problems  and  Exercises 

1.  A  man  bought  a  certain  number  of  ducks,  three  times 
as  many  turkeys,  and  five  times  as  many  chickens  as  turkeys. 
In  all  he  bought  seventy-six  head  of  poultry.  How  many 
ducks  did  he  buy  ?  How  many  turkeys  ?  How  many 
chickens  ? 

2.  A  thermometer  showed  a  certain  number  of  degrees 
rise  one  hovu:,  and  2  degrees  more  than  3  times  as  many  degrees 
rise  the  next  hour.  The  rise  for  both  hours  was  14  degrees. 
Find  the  rise  for  each  hour. 

3.  The  rise  of  a  mercury  column  in  two  hours  amounted 
to  7  degrees,  and  the  rise  the  second  hour  was  3  degrees  less 
than  4  times  the  rise  the  first  hour.  Find  the  rise  for  each 
hour. 

4.  In  two  hours  the  total  rise  of  the  mercury  was  3  degrees, 
and  the  rise  the  second  hour  was  5  degrees  more  than  the 
change  the  first  hour.  Find  the  change  for  each  hour.  What 
do  the  results  mean  on  the  thermometer  ? 

5.  In  two  hours  the  total  rise  of  the  mercury  was  2  degrees 
and  the  rise  the  second  hour  was  8  degrees  more  than  twice 
the  change  the  first  hour.  Find  the  change  in  the  reading 
for  each  hour.  Interpret  the  results  by  means  of  the  ther- 
mometer. 


General  Uses  of  the  Equation                       ii 

The  following  exercises  test  and  review  the  laws  of  arith- 
metical calculation,  and  give  exercise  in  algebraic  equations. 
Find  the  values  of  the  letters,  doing  all  you  can  orally: 

6.  2:x;+i=6  37,  4. 5:!C— 1. 25:^-1-12.7=38. 7 

7.  33P— 2  =  12  38.  i6.5X-l-i5.8— 2.3:x;  =  i86.2 

8-  4^-3  =  12  39-  6.15^-1.65^-1-7.8=57.3 

9-  5^+3  =  15  40.  8^+6.875-^23^=46.875 
.0.  85-7  =  53  41.  z-f5. 372-8.73=61. 34 
II.  8.^-45  =  26  42,  13/— 8.75^-1-6.87=57.87 
[2.  9/-l-ii=92  43.'i5^-f-3.735(;-9.23=65.69 

13.  10/  — 11  =  104  44-  321X— i09X-|-8x=22 

14.  12^4-8=98  45.  404)"— 304 J -|- 12)' =560 

15.  i2/-6  =  74  <    ^    ^                                           f 

16.  126  +  16=82  46.  -+-=3                           .   ^   ^^ 

17.  12^-9=43  ^    ^ 

18.  12^  +  7=95  47-  3~6"'^° 

19.  12^-15=30  t     t 

20.  i6>'  +  i3  =  73  48.  -+-=8 

21.  18)^-12^=33 

22.  212+15  =  102  4p,  1—1==(^ 


23.  252-17  =  113 


4     7 


24.  28:^+14  =  158  50,  5-^  =  7 

25.  28:^—9=251  2     3 

26.  20:!C  +  25(;  — l8x  =  22  s      s 

51. =4 

27.  17)'— 3)'+ 16)'  =  105  46 

28.  175  +  75  —  135=88  i^ 

29.  16^+2/  — 13^  =  22^ 

30.  3.4:^- i.2;x;+4.8x=7o 

31.  3.5:x;+7.6:v— 8.6:)(;=i5  53-  ^~^^ 

32.  5.8>'-3.9)'  +  i2.6)'  =  58 

33-  65-3.55+5.55=68  54.  ^  =  20 

34-  7^-3-5^  +  16.2=54.7  ^ 

35.  6.825  +  1.185-3.55=45  9^^ 

36.  8x— 4. 5:x;+ 5.2:^=87  ^^'  r 


52.  — =5 
^      a     ^ 

4 


12  First- Year  Mathematics 

Problems  on  Percentage  and  Interest 

1.  Find  the  percentage  of  $120  at  4%;    at  6%;   at  7^%; 
atf%. 

2.  Find  the  percentage  at  8%  of  $25;  of  $250;  of  $1,527; 
of  $6. 

3.  Find  the  percentage  at  r%  of  $20;  of  $80;  of  $b. 

4.  Calling  p  the  percentage,  b  the  base,  and  r  the  rate, 
show  that 

P=bX—,  (i) 

100 

and  show  that,  by  one  of  the  laws  of  multiplication  of  frac- 
tions this  may  be  written 

100 

5.  Read  both  (i)  and  (2)  of  problem  4  and  translate  them 
into  words. 

6.  Find  the  interest  on  $175  at  4%  for  2  years;  for  5  years; 
for  f  of  a  year;  for  2 J  years;  for  t  years. 

7.  Find  the  interest  on  $600  for  5  years  at  3%;   at  5%; 
at  8%;  at  6^%;  at  r%. 

8.  Find  the  interest  on  $160  for  /  years  at  6%;    at  3^%- 
at  r%. 

9.  Find  the  interest  at  6%  for  3  years  on  $200;   on  $360; 
on  $756;  on  $/». 

10.  Find  the  interest  at  5%  for  t  years  on  p  dollars. 

11.  Find  the  interest  i,  at  r%  for  t  years  on  p  dollars. 

12.  State  in  words  the  meaning  of 

i=pX—Xt. 
100 

13.  State  in  words  the  meaning  of 

.    pXrXt 

t=- . 

100 


General  Uses  of  the  Egtiation  13 

Important  Conclusions 

1.  An  equation  is  an  expression  of  balance  of  values,  or 
of  numbers. 

2.  The  equation  is  a  convenient  tool  for  problem-solving. 

3.  To  solve  a  problem  by  the  use  of  the  equation,  one 
equation  is  derived  from  another 

(i)  by  adding  the  same  number  to  both  sides, 

(2)  by  subtracting  the  same  number  from  both  sides, 

(3)  by  multiplying  both  sides  by  the  same  number,  or 

(4)  by  dividing  both  sides  by  the  same  number. 

4.  Many  problems  may  be  more  easily  and  simply  solved 
by  the  equation  than  by  arithmetical  methods. 

5.  All  the  arithmetical  processes  of  addition,  subtraction, 
multiplication,  and  division,  of  both  integers  and  fractions, 
must  be  well  known  in  algebra. 

6.  Solving  problems  by  the  equation  sometimes  leads  to 
subtracting  larger  from  smaller  numbers  and  necessitates 
using  negative  numbers,  or  "numbers  below  zero"  on  the 
thermometer  scale. 

7.  Percentage  and  interest  problems  may  be  easily  and 
quickly  solved  by  the  equation. 


CHAPTER  II 

USES  OF  THE  EQUATION  WITH  PERIMETERS  AND  AREAS 

Indicating  Arithmetical  Operations  Algebraically 

1.  A  boy  rides  on  his  bicycle  8  mi.  in  one  hour  and  5  mi. 
the  next  hour;  how  far  does  he  ride  in  the  two  hours? 

Note. — Write  the  sum  of  8  and  5,  not  13,  but  8+5.  The 
form  8+5  is  just  as  truly  a  sum  as  is  13.  It  will  sometimes 
be  desirable  to  refer  to  the  form  8  +  5  as  the  indicated  sum. 

2.  If  the  boy  rides  a  mi.  the  first,  and  7  mi.  the  second 
hour,  how  far  does  he  ride  in  the  two  hours  ? 

3.  A  boy  has  m  marbles  and  buys  p  more;  how  many  has 
he  then  ? 

4.  A  boy  had  18  marbles  and  lost  7 ;  how  many  had  he  then  ? 

Note. — Give  the  indicated  diference. 

5.  A  boy  had  m  marbles  and  lost  8  of  them;  how  many 
had  he  left  ? 

6.  A  boy  had  m  marbles  and  lost  n  of  them;  how  many 
had  he  left  ? 

7.  Show  the  sums  of  these  pairs  of  numbers  and  the  differ- 
ences, the  first  of  the  given  numbers  being  the  minuend,  and 
the  second,  the  subtrahend: 

(i)  :»  and    7  (5)     5  and  /  (9)  :^  and  a 

(2)  a    "    12  (6)  15    "    n  (10)  d    ''     c 

(3)  y    "    10  (7)     r    "    s  (II)  c    "     b 

(4)  X    "      y  (8)    a    "    X  (12)    t    "    m. 

8.  How  many  yards  of  cloth  are  12  yd.  and  10  yd.  ? 

9.  How  many  dozens  of  eggs  are  8  doz.  and  4  doz.  ? 

10.  How  many  12 's  are  5  12's  and  4  12's? 

11.  How  many  times  12  are  9X12  and  4X12  ? 

14 


Uses  of  the  Equation  15 

12.  How  many  half-dozens  are  8  half-dozens  and  6  half- 
dozens  ? 

13.  How  many  times  6  are  8X6  and  6X6  ? 

14.  How  many  times  3  are  5X3  and  7X3  and  10X3  ? 

15.  How  many  times  x  are  2  times  x  and  3  times  x?  x  and 
X?  X  and  3  times  x ? 

5.  It  is  customary  in  algebra  to  write  x+x,  x+x-\-x, 
x-\-x-\-x+x,  etc.,  thus,  231;,  35c,  /^x,  etc.,  and  to  read  them 
"two  ac,"  "three  5C,"  "four  x,^\  and  so  forth. 

But  X  times  x,  x  times  x  times  x,  and  a;  times  x  times  ^c 
times  X  are  written  ac^,  aj3,  x*,  etc.,  and  are  read,  "a;  square," 
"rx;  cube,"  ":x;  fourth  power,"  etc. 

X  square  may  be  read  '^x  2d  power"  or  "x  2d;"  x  cube 
may  be  read  "jc  3d  power,"  or  ^'x  3d;"  and  ''x  fourth 
power"  may  be  called  ":r  4th,"  etc. 

The  dot  is  often  used  instead  of  the  times  sign  to  indicate 
multiplication.  Thus,  ^cX^X^  may  be  written  x-x-x, 
meaning  x  times  x  times  x. 

6.  In  2X,  ^x,  4X,  ax,  etc.,  the  2,  3,  4,  a,  etc.,  are  called 
coefficients  of  x. 

7.  In  X',  x^,  x^,  x^,  etc.,  the  2,  3,  4,  w,  etc.,  are  called 
exponents  of  x. 

What  would  be  the  written  form  of  ''x  5th?"  "ic  6th?" 
"^  7th?"  "3C  loth?"  ''x  »th?"  What  would  be  the  mean- 
ing of  each  of  these  forms  ? 

8.  Notice  that  4a;  means  4  •  ic,  or  that  x  is  to  be  used  as 
an  addend  4  times,  while  x*  means  x  •  x  •  x  •  x,  or  that  x  is 
to  be  used  as  a  factor  4  times,  and  similarly,  for  the  other 
forms,  as  2,x  and  x^,  and  ^x  and  x^,  etc. 

Letting  x=(i,  give  the  values  of 


I.    2X 

4.    iC3 

7-  S^ 

10.  33?" 

2.  ac" 

5.    43c 

8.  x^ 

II.  5^3 

3-  2>x 

6.    5C4 

9.  2:x;^ 

12.    2X^, 

i6 


First-Year  Mathematics 


13.  Describe  briefly  the  meaning  of  factor  and  of  addend 
as  they  are  used  in  arithmetic. 


.2* 


Fig.  5 


Expressing  Perimeters  and  Areas 

I.  What  is  the  sum  of  the  sides  of  a  triangle 
(Fig.  5)  whose  sides  are  2x  ft.,  2X  ft.,  and  ^x 
ft.  long  ?  Express  the  sum  as  a  certain  num- 
ber of  times  x. 

2.  What  is  the  sum  of  the  three  sides  2a,  5a,  and  6a  of  a 
triangle  ? 

3.  A  lot  has  the  form  of  an  equal-sided 
(equilateral)  triangle  (Fig.  6),  each  side  being  x 
rd.  long.  How  many  rods  of  fence  will  be  needed 
to  inclose  it  ? 

9.  Any  figure  whose  sides  are  all  equal  is  called 
an  equilateral  figure;  as  equilateral  triangle,  equilateral  penta- 
gon, etc. 

The  sum  of  all  the  sides  of  any  closed  figure  is  called  the 
perimeter  of  the  figure. 

y  I.  In  Fig.  7,  called   a  parallelogram,  what 

jg/  X  part  of  the  perimeter  does  x+y  equal?    What 


Fig.  6 


y 

Fig.  7 


is  the  whole  perimeter? 

2.  Show  (i)  that  the  perimeter,  p,  of  a  square 


10 


whose  side  is  8  is  given  by  ^=4  •  8  and  (2)  that  the  area,  A, 
is  given  by  A  =8^. 

3.  Show   by  equations     2 
the  perimeter,  p,  and   the 
area,  .4,  of  a  square  whose 
side  is 
(i)  12    (4)  a     (7)  IO-I-2 

(2)  s      (5)  y     (8)  6-H7 

(3)  X     (6)  26    (9)  c+d. 

4.  What  is  the  perimeter  of  a  square  of  which  the  area  is 
16  sq.  ft.  ?    a^  sq.  ft.  ?  40^  sq.  ft.  ?  x^  sq.  ft.  ? 


■(2 


10 

Fig.  8 


Fig.  9 


Uses  oj  the  Equation  17 

5.  What  is  the  area  of  a  square  whose  perimeter  is  20  ft.  ? 
4^  ft.  ?    8a  ft.  ? 

6.  Draw  figures  to  illustrate  your  solutions  of  5. 

7.  The  dimensions  of  a  rectangle  are  a  and  b,  what  is  the 
perimeter?  The  half-p eri meter ?  The  sum  of  a  pair  of  op- 
posite sides  ?    The  sum  of  the  other  pair? 

8.  What  is  the  area  of  the  rectangle  of  problem  7  ? 

10.  Numbers  denoted  by  letters  are  literal  numbers.  The 
product  of  two  different  literal  numbers,  as  x  and  y,  is  shown 
by  writing  the  letters,  or  factors,  side  by  side,  as  xy,  with  no 
sign  between.  We  are  familiar  with  the  form  xXy  from 
arithmetic.  The  form  xy  is  most  used  in  algebra.  It  is  often 
convenient  to  use  the  form  x  •  y. 

1.  Recalling  that  the  exponent  of  a  number  shows  how 
many  times  the  number  is  to  be  used  as  a  factor,  give  the 
products  of  the  following  pairs  of  factors: 

(i)  4'  '  4'  (5)  ^'  •  «  (9)  b'  •  b' 

(2)  8  •  83  (6)  a3  •  a3  (10)  c  •  c^ 

(3)  10^  •  ic*  (7)  a  •  fls  (11)  c3-c^ 

(4)  123  .  124  (8)  ^4  .  a^  (12)  x^  -  x^. 

2.  Show  from  Fig.  10,  (i)  that  the  per- 
imeter, p,  of  a  square  of  3X  is  given  by 
P  =  i2x,  and  (2)  that  the  area,  A,  is  given  by 

A  =QX'. 

X 

3.  Express  by  an  equation  the  area.  A,         x     x     x 
of  a  rectangle  that  is  8  in.  long  and  5  in.  Fto.  10 
wide;  of  the  same  length  and  4  in.  wide;    3^  in.  wide;  6J  in. 
wide. 

4.  Express  by  an  equation  the  area,  ^,  of  a  rectangle  12  in. 
long  and  of  the  following  widths:  6  in.;  8}  in.;  9 J  in.;  lof 
in.;  xm.',  y  in. 


'i    i 

1           1 

r       r 

1 
1       1 

i8 


First-Year  Mathematics 


5.  Express  by  equations  the  areas  of  rectangles  /  in.  long 
and  of  the  following  widths:  12  in.;  gin.;  ^in.;  nin.;  x'm.; 
a  in. 

6.  Express  by  equations  the  areas  of  rectangles  of 
width  w  and  of  the  following  lengths:  8;  10;  12J;  x;  a; 
I;  h;  z. 

7.  Express  by  equations  the  products  of  the  following 
pairs  of  factors: 

(5)  h   by  h^  (9)  x^  by  x 

(6)  a^  by  a  (10)  a    by  rc» 

(7)  a*  by  a3  (11)  a'  by  ap 

(8)  X  by  3c3  (12)  g  by  <». 

8.  Show   from   Fig.    ii,   that   the  area, 
^,  of    a  rectangle    2X  by  3a?  is  given    by 

9.  Show  that  the  area,  ^,  of  a  rectangle 
^a  by  5a  is  given  by  A  =  i^a'. 

10.  Express   by   an    equation   the 
area,  A,  of  parallelograms  of  altitude    i    ►' 
a,  and  of  the  following  lengths:    10;    1    /  "*■ 

12^;    16.8;    h;  c;  x;  2»;  53. 


(I) 

ahy  X 

(2) 

6  by  c 

(3) 

6by& 

(4) 

a  by  a 

^! 

1 

j 

or    x     ac 

Fig. 

ir 

b 

II.  Express   by   an    equation    the  ^°"  ^^ 

7   area,  ^,  of  triangles  of  altitude  a,  and  of 
<jfL     x^^^      /      the  bases,  12;    18 J;    20.25;   6;  i;  x;   2y; 
42;  5^- 


6 

^°-  ^3  12.  Show    that    the    area.     A,    of    the 

trapezoid,  in  the  accompanying  Fig.  14, ^ 

6+8 


is  given  by  vl  =4 

13.  Show   that   the   area,   A,  of   the 


Fig.  14 


Uses  of  the  Eqiiation 


19 


trapezoid,  in  the  accompanying  Fig. 

9  +  7 
2 


15,  is  given  by  ^=3 


i  - 


9 

Fig.  15 


14.  Show  that  the  line  EF=i(&+c), 

p  meaning  ^  of  the  sum  of  b  and  c  of  Fig.  16. 

15.  Show    from     Fig.    16    that    the 
area,   A,   of   the  trapezoid   is  given   by 
b-\-c    a{b-{-c) 
2  2 


A=a 


(1) 


(4-) 

(3) 


16.  Express  by  an  equation  the  area,  .4,  of  a  rectangle  of 
dimensions 

(i)  6  and  ic+3  (3)  a  and  re— 2 

(2)  acand  a+5  (4)  a+2  and  x-\-j. 

17.  Indicate  the  area  of  a  rectangle  of  dimensions  a-\-b 
and  x-f  >*. 

Note. — ^The    product    of    w+«    and    c-\-d    is    written 
(m+»)(c+d). 

18.  Express  in  terms  of  its  base 
and  altitude  the  area  of  the  rect- 
angle (i)  of  Fig.  17;  of  (2);  of  (3); 
of  (4). 

19.  How   then   may   you   express 
(m+n)  (c+d)  by  using  the  truth  that  any  whole  equals  the 
sum  of  all  its  parts  ? 

20.  State  in  words  the  value  of 

(i)  (x+y)ia+b)  (3)  (a+xXb+y) 

(2)  (x+y){m-\-n)  (4)  (r+sXa+x). 

21.  Show  by  a  figure  the  value  of 

(i)  (a+bXa+b)  or  (a+by  (3)  (x+y)' 

(2)  (c+d)'  (4)  (m+n)'. 


Fig.  17 


20 


First-Year  Mathematics 


22.  Each  of  the  following  expressions  is  the  product  of 
what  two  equal  numbers  ? 

(i)  a''-\-2ax-\-x^  (3)  k^-\-2kh-irh''  (5)  x''->r()X-\-g 

(2)  h^-\-2bc  +c'  (4)  s'  +  2sl  +t'  (6)  c'+Sc+ib. 

23.  The  base  of  a  rectangle  is  8  yd.  and  the  area  is  40 
sq.  yd.     What  is  the  altitude,  a4 

24.  What  is  the  altitude,  a,  of  a  rectangle  having — 

a  base =8  in.  and  an  area  =32  sq.  in.  ? 
"  =8  in.  "  "  =i6sq.  in.  ? 
"  =8  in.  "  "  =i2sq.  in.? 
«     =5  ft.     "         "       =7i  sq.ft.? 

25.  Show  by  an  equation  the  base,  b,  of  a  rectangle  having — 

an  altitude =9  ft.  and  an  area  =  27  sq.  ft. 
=9  ft.  "  "  =18  sq.ft. 
=9  ft.  "  "  =15  sq.ft. 
=9  ft.  "  "  =12  sq.ft. 
=9  ft.       "       "        =A  sq.  ft. 

26.  Show  by  an  equation  the  other  dimension  of  a  triangle 
having — 

a  base =6  ft.  and  an  area =24  sq.  ft. 
"  =6  ft.  "  "  =12  sq.ft. 
•'  =6  ft.  "  "  =9  sq.  ft. 
"     =6  ft.       "       "        =3  sq.  ft. 

an  altitude  =4  yd.  and  an  area  =  16  sq.  yd. 


=4  yd.     " 

"      =8  sq.  yd 

=4  yd.       " 

"      =4  sq.  yd 

=a  ft. 

"      =  a  sq.  ft. 

=h  rd.       " 

"      =}il  sq.  ft. 

=i  in.        " 

"      =ab  sq.  in. 

=b  in.       " 

"      =by  sq.  in. 

=c  in.       " 

"      =  a  sq.  in. 

=a  in.       " 

"      =  b  sq.  in. 

Uses  of  the  Equation  21 

X 

11.  The  Quotient  of  x  divided  by  y  is  written  - ,  or  x-^r 

y  ' 

and  is  read  "rv  divided  by  >»."    -  is  also  read  "x  over  y" 

Show  by  an  equation  the  quotient,  q,  of  the  first  of  the.  c 
numbers  divided  by  the  second: 

1.  m     and  n       5.  a-\-b      and  c  9.  a'—b'    and  a—d 

2.  a        "    b       6.  a  "   x-j-y       10.  x'  "    a  +  b 

3.  6         "a       7.  a+b         "    c-\-d       11.  (a+b)'    "    a  +  b 

4.  4X       "    sy     8.  a'-b^*    "    a  +  6        12.  (a-6)»    "    a-b 

*NoTE.  a'—b'  is  read  "a  square  minus  &  square;'" 
(a— by  is  read  "the  square  of  a—b,"  and  (a+&)^  is  read 
"the  square  of  a +  6." 

Definitions 

12.  In  an  expression  Uke  ^xy  +  'jax—gb—4,  the  numbers 
^xy,  "jax,  gb,  and  4,  that  are  connected  by  addition  or  subtrac- 
tion, are  called  terms  of  the  expression. 

13.  A  one-term  expression,  like  8,  "jx,  abc',  etc.,  is  called 
a  monomial. 

14.  The  coeflicient  of  any  factor,  or  number  in  a  term  is 

the  product  of  all  the  other  factors  of  the  term.      Thus,  in 

^axy,  the  coefficient  of  axy  is  4,  of  xy  is  40,  of  ax  is  4^;    in 

be   ,  _  .         .     .   b      . ,  .    c 

—  the  coefficient  of  c  is  -  ,  of  0  is  -  ,  etc. 

3  3  3 

When  the  coefficient  of  the  term  is  spoken  of  the  arith- 
metical factor  is  usually  meant.     Thus,  it  is  common  to  say 

the  coefficient  of  the  term  4axy  is  4;    of  —  is  ^,  etc.    If  no 

coefficient  is  written  as  in  a,  x^,  1   is  understood,  as  though 
I  a,  IX*  were  written. 

15.  The  exponent  of  a  number  is  the  small  figure  or  letter 
written  to  the  right  and  a  little  above  the  number  symbol,  to 


22 


First-Year  Mathematics 


denote  the  number  of  equal  factors  in  a  product.  Thus,  in 
63,  meaning  the  product  6X6X6,  the  3  is  the  exponent.  The 
number  6,  itself,  is  called  the  base,  and  the  product  6^  is  the 
power.  Thus,  216  (=6^)  is  the  3d  power  of  6,  because  using 
6  as  a  factor  3  times  gives  216,  and  so  with  other  numbers. 

When  no  exponent  is  written,  as  in  ax,  the  exponent  is 
understood  to  be  i  for  each  letter,  as  though  the  number  were 
written  a^  x^, 

16.  A  number  as  4ax-\-io,  denoted  by  an  expression  of 
two  terms  is  a  binomial;  a  three-term  expression  is  a  trinomial. 

17.  Number  expressions  consisting  of  two,  three,  four,  etc., 
terms,  are  called  polynomials. 

The  area  of  a  rectangle  whose  altitude  is  5  and  whose  base 
is  the  sum  of  two  Unes,  one  a  inches  and  the  other  b  inches 
long,  may  be  written  thus:  $(a  +  b).  If  the  altitude  were  7 
and  the  base  equal  to  the  difference,  a—b,  the  area  would 
be  written  7  (a— ft). 

18.  The  sign,  (  ),  called  the  parenthesis,  means  that  the 
terms  within,  as  the  a  and  the  b  in  17,  are  first  to  be  added, 
or  subtracted,  and  then  the  sum  or  difference  is  to  be  multiplied 

Give  the  meaning  of  the  following: 

1.  a3  4.  8(40+2)  7.  i6(a+6) 

2.  4a'b  5.  9(60—3)  8.  2o(x—y-{-i) 

3.  6(10-3)  6.  7(80  +  1)  9.  2s{x-a). 

Finding  Values  of  Expressions 

Of  what  kind  of  figures  may  the  following  equations  ex 
press  the  perimeters,  p  ? 


Fig.  18  Fig.  19 


Uses  oj  the  Equation  23 


I. 

P=3X 

5- 

p='jx 

9- 

p  =  l2X 

2. 

p=4X 

6. 

p=8x 

10. 

P=20X 

3- 

p=SX 

7- 

p=gx 

II. 

p=nx 

4. 

p=6x 

8. 

p=iox 

12. 

p=ax. 

13    Give  the  coefficient  of  x  in  the  second 
members  of  problems  i  to  12.  x 


Show,  by  a  sketch,  figures  whose  peri-        x       x     % 
meters,  p,  are  given  by  ^^'  ^° 

14.  p=2{2X-\-2)         17.  p=4X-{-i6  20.  P=zx+zy 

15    ^=2(3:x;+2)         18.  p=6x—i2  21.  p=4X-\-2y. 

16.  p=6x-\-4  19.  ;^  =  i2:v— 24         22.  p=SX-\-;^y. 

23.  Find  the  values  of  the  perimeters  in  problems  i  to  12 
for  x=2  in.;  for  x=^  ft.;  for  x=6  yd. 

24.  Find  the  values  of  the  perimeters,  p,  of  problems  14 
1  to  22,  for  jc=5  and  y=2;  for  ic.=  i2  and  >'=4; 

for  3(;=a  and  y=h. 


■■» — Mi--*  Of  what  plane  figures   do  the  following 
Fig.  21        equations  give  the  areas,  A  ? 

25.  A=x'  30.  A=a{x-{-4)                       4(a-f6) 

26.  ^=3031;  31.  ^=iic(3'— 2)               '               2 

27.  ^=2(:c+i)  32.  ^=>'(x+7)                   j_    /&±f\ 

28.  ^=5(^+3)  ^^3(^+1)            ^^'  ^~\2    ) 

29.  A=x{a-\-2)  2                  36.  ^=:!C*— 4a;. 

37.  Find  the  value  of  A  in  problems  25  to  36  for  x=^,  y=2, 
a=4,  6  =  1,  c  =  i;  for  5f=5,  >'=3,  a=6,  6=2,  c=4. 

19.  An  expression  hke  (fl+S)i.x~y)  nieans  that  a  and  5 
are  first  to  be  added,  then  y  is  to  be  subtracted  from  x,  and 
the  sum,  a +5,  is  to  be  multiplied  by  the  difference,  x—y. 

Give  the  meanings  of  the  following: 

1.  {x+y)(a-{-io)  3.  {x-iry){a—'b) 

2.  [a-\-h){c-\-cl)  4.  (a-&)(c-<f). 


24 


First-Year  Mathematics 


Sketch  and  show  dimensions  of  rectangles  whose  areas  are 
given  by 


X  3 

Fig.  22 


1  « 

a 

n 

0 

S- 

A  = 

(fl  +  i)(x+3) 

6. 

A  = 

(a+5)(^+4) 

7- 

A  = 

{a-2){x+i) 

8. 

A  = 

(a-3)(x+6) 

9- 

A  = 

{a-x){b  +  i) 

lO. 

A  = 

{a-2)ih+2) 

II, 

A  = 

(a-i)(6-i) 

12. 

A  = 

(a-i)(a  +  i) 

13- 

A  = 

(a  +  i)(a  +  i) 

14. 

A  = 

(a  +  2)(o+4) 

15- 

A  = 

(a-4)(a+4) 

16. 

A  = 

(a+6)(a+6). 

a?  1 

Fig.  23 

20.  The  signs,  V      and  #^     ,  called  radical  signs,  have 
the  same  meaning  as  in  arithmetic. 

Give  the  values  of  the  following  for  a  =3,  6=3,  c  =  i,  /=8, 
x=io: 

6.  3x(cJ+8-5c) 

7.  (a+c)(:x;-/) 

8.  a  +  6-f/ 

9.  Va' 
10.  l/a6 


1.  (a  +  &)(c  +  i) 

2.  8(a+x) 

3.  a(6+<;) 

4.  t{ab—c) 

5.  a(:x;+&— c) 


II 


l/6< 

i\'t 

13.  :)C* 

14.  25-T3 


The  Use  of  the  Fundamental  Laws  in  Equations 

I.  The  width  of  a  rectangle  is  x—t,,  the  length  x,  and 
the  perimeter  is  66  yards.  Find  the  width  and  the  length 
in  vards. 


4«  —  6  =  66. 

(I) 

;  ? 

(2) 

1  ^ 

4X  =  J2. 

jr 

•O 

»  =  i8. 

(3) 

^ 

4 

»-3  =  i5- 

(4) 

a? 

Fig.  24 

Z75g5  of  the  Equation  25 

The  perimeter  being  denoted  by  both  66  yards  and  4«— 6,  we 
write  the  equation  ♦  ♦• 

Adding  6  to  both  sides 

Dividing  both  sides  by  4 

Subtracting  3  from  both  sides 

The  width  is  15  yards  and  the  length  18  yards. 
Check:    15  +  18  +  15  +  18  =  66,  or  from  equation  (i)  4»i8  — 6  =  72  — 6 
=66. 

2.  The  width  of  a  rectangle  is  x  ft.,  the  length  3C  +  12  ft., 
and  the  perimeter  is  144  ft.     Find  the  width  and  length. 

Both  144  and  4;e  +  24  denoting  the  perimeter,  we  write 

4«+24  =  i44.  (i) 

Subtracting  24 

4a;  =  i2o.  (2) 

Dividing  by  4 

X—  30,  the  width.  (3) 

Adding  12 

ic  +  i2=  42,  the  length.  (4) 

Check:  30  +  42  +  30  +  42  =  144,  or  from  equation  (i)  45^+24=4  .  30  +  24 
=  120  +  24  =  144. 

3.  One-fifth  of  the  perimeter  of  a  field  plus  3^  rods  is  10  rods. 
Find  the  perimeter. 

Call  p  the  perimeter,  then 

^+3i  =  io.  (I) 

Multiply  both  sides  by  10 

2p  +  3S  =  ioo.  (2) 

Subtract  35 

2p=  65.  (3) 

Divided  by  2 

P=  32i.  (4) 

Check:  —+3i=6i +  3^  =  10. 

4.  In  the  solution  of  problem  i  show  where  each  of  the 
following  laws  is  used : 


a6  First-Year  Mathematics 

(i)  If  the  same  number  be  added  to  equal  numbers,  the 
sums  are  equal; 

(2)  If  equal  numbers  be  divided  by  the  same  number,  the 
quotients  are  equal; 

(3)  If  the  same  number  be  subtracted  from  equal  numbers, 
the  remainders  are  equal. 

5.  Show  where  the  laws  of  problem  4  are  used  in  the  solu- 
tion of  problem  2. 

6.  In  the  solution  of  problem  3,  show  where  the  law  is 
used:  "if  equal  numbers  be  multiplied  by  the  same  number 
the  products  are  equal. 

7.  In  the  same  solution  show  where  the  following  are  used : 
"If  the  same  number  be  subtracted  from  equals  the  re- 
mainders are  equal." 

"If  equal  numbers  be  divided  by  the  same  number  the 
quotients  are  equal." 

21.  The  four  laws  thus  far  used  are 

I.  //  the  same  number  be  added  to  equals  the  sums  are  equal, 

II.  //  the  same  number  be  subtracted  from  equals  the  re- 
mainders are  equal, 

III.  7/  equal  numbers  be  multiplied  hy  the  same  number, 
the  products  are  equal, 

IV.  //  equal  numbers  be  divided  by  the  same  number,  the 
quotients  are  equ^. 

Solve  the  following,  using  the  equation  and  showing  which 
fundamental  law  is  used  in  each  step. 

1.  The  perimeter  of  an  equilateral  triangle  is  27  rods.  Find 
a  side. 

2.  The  perimeter  of  an  equilateral  quadrilateral  is  48  ft. 
Find  a  side. 

3.  The  perimeter  of  an  eqiulateral  hexagon  (6-side)  is  186. 
Find  a  side. 


Uses  of  the  Equation  27 

4.  The  perimeter  of  an  equilateral  decagon  (lo-side)  is  255. 
Find  a  side. 

5.  The  perimeter  of  an  equilateral  dodecagon  (12-side)  is 
294.     Find  a  side. 

6.  The  area  of  a  rectangle  is  96  yards,  the  base  is  ic+8 
and  the  altitude  is  8.     Find  the  base.     Find  x. 

7.  The  altitudes,  bases,  and  areas  of  parallelograms  are  as 
given.  Find  the  value  of  the  unknown  dimension  in  arith- 
metical numbers. 

Altitudes  Bases  Areas 

x-s  4  12 

SX+2  8  56 

10  3^+2  no 

16  5^—7  28S 

2:x;+i  15  105 

25  60a +40  2,500 

283^—17  30  840 

Application  of  the  Fundamental  Laws 

22.  The  four  fundamental  laws  stated  above  may  be  re- 
ferred to  thus: 

I  is  the  addition -law,  II,  the  subtraction -law,  III,  the 
multiplication- law,  and  IV,  the  division-law.  These  laws 
are  also  called  axioms. 

Exercise  IV 

Solve  the  following  equations,  i.  e.,  find  the  number  for 
which  the  letter  stands,  and  show  where  each  of  the  four  laws 
just  stated  is  used :  , 

1.  ic— 5  =  12    ■•  ■  6.  3^—4  =  14  II.  i2:x;=72 

2.  2:x;— 6  =  18  7.  4^  +  1=21  12.  ii:j;=9o 

3.  3X+i  =  i9  8.  45(;— 6  =  18  13.  |:x;— 2  =  10 

4.  3:»— 1=20  9.  ^x+2=2(i  14.  f:x; -1-14=20 

5.  35e-f4=25  10.  45i;-i6=32  15.  |5(;-io=2 


28  First- Year  Mathematics 

i6.  Y+3  =  i7  «•  -+ii=Si  27.  ^A:Y-^=2* 


2a 
•7. --.-5 


7^  ■ 

o  28.    — I —  =  2 

8x  ,  25 


18.  2x— 3^=3^  24.  - — 1-2=26 

7  oc    00 

19.  4^  +  2^=31  2(jC  +  2)       _  29.    ---=3 

20.  47-2^=131  '5.  ^— -«  ^     5 

21. *=3*  26.  ^^ ^-=3  30.  --i  =  x. 

3  ^  3  78 

Summary 
Chapter  II  has  exemplified  or  taught  the  following: 

1.  Numbers  may  be  denoted  by  letters.  Numbers  thus  de- 
noted are  called  literal  numbers, 

2.  A  sum  is  indicated  by  writing  the  pliis  (+)  sign  between 
the  numbers  to  be  added. 

3.  A  difference  is  shown  as  in  arithmetic  by  writing  the 
minus  (  — )  sign  between  the  minuend  and  the  subtrahend. 

4.  A  product  of  Hteral  numbers,  or  of  arithmetical  num- 
bers by  literal  numbers,  is  usually  shown  by  writing  the  factors 
side  by  side  with  no  sign  between  them.  The  multiplication 
dot  ( • )  and  the  times  (X)  sign  may  also  be  used  between 
factors  to  indicate  their  product. 

5.  A  quotient  is  indicated  as  in  arithmetic,  or  by  means  of 
the  fractional  form,  the  dividend  being  written  above  and  the 
divisor  below  a  horizontal  Une. 

6.  Powers  are  denoted  by  writing  small  figures,  or  letters 
to  the  right  and  above  the  base,  the  number  of  units  in  the 
figure,  or  letter,  showing  how  many  times  the  base  is  a  factor 
in  the  power. 

7.  Roots  are  shown  by  the  radical  sign,  as  in  arithmetic. 


Uses  of  the  Equation  29 

8.  An  equal-sided  figure  is  an  equilateral  figure. 

9.  How  to  denote  perimeters  and  areas  of  geometrical 
figures  by  algebraic  expressions,  and  equations. 

10.  How  to  find  the  product  of  binomials  by  monomials, 
and  of  two  binomial  sums,  both  algebraically  and  geometri- 
cally. 

11.  A  single  number,  or  a  product  or  quotient  of  numbers 
is  a  term. 

12.  A  one-term  number  is  a  monomial ;  a  two-term  number 
is  a  binomial ;  a  three-term  number  is  a  trinomial ;  and  a  num- 
ber expressed  by  any  number  0}  terms  is  a  polynomial. 

13.  The  coefficient  of  any  factor  in  a  term  is  the  product 
of  all  other  factors  of  the  term. 

14.  The  coeflScient  0}  a  term  usually  means  the  arithmeti- 
cal factor  of  the  term. 

15.  The  exponent  of  a  power  is  the  number  that  shows 
how  many  times  the  base  is  a  factor  in  the  power. 

16.  The  value  of  a  letter  is  the  number  for  which  the  letter 
stands. 

17.  To  find  the  value  of  an  expression  means  to  substitute 
values  for  the  letters,  and  to  reduce  the  result  of  all  operations 
indicated  to  the  simplest  form. 

18.  Changes  in  an  equation  can  be  made  only  in  accordance 
with  a  few  fundamental  laws,  called  axioms. 


CHAPTER  III 


THE  EQUATION  APPLIED  TO  ANGLES 

The  Measurement  of  Angles 

23.  What  part  of  a  complete  turn  does  the  hand  of  a 
clock  or  watch  make  (Fig.  25)  when 
it  rotates  (turns)  about  the  center 
post  from  12  to  3  ?  12  to  6?  12  to 
9?  12  to  4?  12  to  8?  12  to  2? 
12  to  10? 

24.  If  a  line,  OA  (Figures  26, 

27,  28,  29),  be  imagined  to  rotate 

(turn)   in  a  plane    about  a  fixed 

_,  point,  O,  in  the  direction  indicated 

Fig.  25  I-        >      > 

by  the  arrow-heads,  until  it  reaches 
the  position  O  B,  it  is  said  to  turn  through  the  angle  x  (also 


Fig.  26 


Fig,  28 


Fig.  29 


30 


The  Equation  Applied  to  Angles 


31 


Fig.  30 


called  the  angle  A  OB).     In  Fig.  30  the  rotating  line  has 
made    a    complete    turn    about    the 
point   O. 

The  word  "angle"  means  "the  amount 
of  turning"  of  a  rotating  line.  The  original 
meaning  of  the  word  was  "corner,"  from 
the  Latin  angulus. 

Draw  an  angle  made  by  a  line  which  has  rotated  ^  of  a 
complete  turn;  ^  of  a  complete  turn;  f  of  a  complete  turn; 
I J  turns;  if  turns. 

25.  If  a  line,  rotating  in  a  plane  about  one  of  its  points, 
makes  i  of  a  complete  turn,  the  angle  it  makes  is  called  a 
right  angl«»  (Fig.  31);  if  the  Hne  makes  a  half -turn  the  angle 
is  called  a  straight  angle  (Fig.  32). 


X 


0  ^A 

Fig.  31 

Draw  an  angle  equal 


(I 
(2 
(3 
(4 
(5 
(6 
(7 
(8 
(9 
(10 


B  0  ~A 

Fig.  32 


to  two  right  angles 

to  three  right  angles 

to  four  right  angles 

to  five  right  angles 

to  two  and  a  half  right  angles 

to  two  straight  angles 

to  one  and  a  half  straight  angles 

to  two  and  a  half  straight  angles 

to  three-fourths  of  a  straight  angle 

to  one  and  three-fourths  straight  angles. 


2.  How  many  right  angles  are  there  in  an  angle  made  by 


32 


First-  Year  Mathematics 


a  complete  turn  of  a  rotating  line  ?    By  a  half  turn  ?    By  7 
turns?     By  i\  turns?      By  /  turns? 
5^+7 


By  -  turns  ?      By  ^ 
■'2  '4 


turns  ?    Bv 


turns  ? 


3.  How  many  straight  angles  are  there  in  an  angle  made 
by  a  complete  turn  of  a  rotating  line  ?    By  i  J  turns  ?    By  |  of 

7t 
a  turn  ?  By  ^  of  a  turn  ?  By  9  turns  ?  By  3/  turns  ?  By  —  turns  ? 

4 
By  3(2/— 5)  turns? 

4.  How  many  right  angles  are  there  in  a  straight  angle? 
In  5  straight  angles?     In  7^  straight  angles?     In  5  straight 

5  I  "iS 

angles  ?    In  -  straight  angles  ?    In  -~-  straight  angles  ?    In 
6  8 

95+8  straight  angles?    In  35—2  straight  angles? 

5.  How  many  straight  angles  are  there  in  4  right  angles? 
In  2  right  angles  ?   In  3  right  angles  ?    In  7  right  angles  ?    In 

r  right  angles  ?     In  —  right  angles  ?     In  —  right  angles  ? 

In  6r+8  right  angles  ?    In  4^—6  right  angles  ? 

6.  Fold  a  crease  (A  O  B,  Fig.  ;^^)  in 
a  piece  of  paper.  Fold  again  so  that 
the  edge,  O  A,  falls  along  the  edge,  O  B. 
Unfold  the  paper.  The  two  creases, 
A  O  B  and  C  O  D  (Fig.  34),  form  four 
angles,  x,  y,  z,  w.  Show  that  these 
angles  are  equal.  Are  they  right  angles  ? 
Why? 

26.  On  the  protractor  (angle  meas- 
urer) shown  in  Fig.  35,  the  right 
angle,  XOY,  is  divided  into  90  equal 
angles.  Each  of  these  angles  is  a 
degree. 


The  Equation  Applied  to  Angles 


33 


35  degrees  is  written  briefly,  35°.     The  symbol  for  angle 
is  Z ;  for  angles,  is  Z^. 

I.  How  many  degrees  are  there  in  ZX  O  A  (Fig.  35)  ?    In 
ZXOB?    InZAOB?    InZXOC?    InZAOC? 


Fig.  35 


2.  Express  by  an  equation  the  number  of  degrees  in 
ZXOA+ZAOB  (Fig.  35);  in  ZAOB  +  ZBOY;  in 
ZXOB-ZAOB;  in  ZXOB-ZXOA. 

3.  How  many  degrees  are  there  in  2  straight  angles  ?    In 

4  right  angles  ?    In  §  of  a  right  angle  ?    In  |  of  a  right  angle  ? 

'ir 
In  -^  of  a  right  angle  ?    In  r  right  angles  ?    In  -^  right  angles  ? 

lo 

In  f +7  right  angles  ?    In  — -  right  angles  ? 


27.  The  protractor  is  commonly  a  semicircle  (Fig.  36). 
If  a  straight  line  were  drawn  from  each  mark  on  the  circular 


34 


First-Year  Mathematics 


rim  to  the  center,  O,  i8o  equal  angles  would  be  formed  at 
O  each  of  which  would  be  an.  angle  of  one  degree  (i°). 


Fig.  36 

1.  Suppose  a  pointer  to  rotate  about  the  fixed  point,  O 
(Fig.  36),  in  the  direction  indicated  by  the  arrow.  In  rotating 
from  the  position  O  X  to  the  position  O  Z  through  how  many 
right  angles  would  it  turn  ?  Through  how  many  straight 
angles  ?    Through  how  many  degrees  ? 

2.  Find  the  number  of  degrees  and  the  number  of  right 
angles  in  each  of  the  following  angles  of  Fig.  37:  XOA; 
XOB;  XOY;  X  O  C;  XOD;  AOB;  A  O  Y;  B  O  C; 
BOD;   COZ;   A  O  Z;   X  O  Z. 

3.  Read  the  angle  in  Fig.  37  which  is  the  sum  of  angles 
X  O  A  and  A  O  B;  of  angles  A  O  B  and  B  O  C;  of  angles 
X  O  B  and  B  O  Y. 

4.  Angle  X  O  B  is  the  sum  of  what  two  angles  of  Fig.  37  ? 
Answer  by  an  equation. 

5.  Show  by  an  equation  that  the  sum  of  two  angles  of 
Fig.  37  equals  ZA  O  C;    ZX  O  C. 


The  Equation  Applied  to  Angles 


35 


6.  Show  by  an  equation  that  the  sum  of  three  angles  of 
Fig.  37  equals  ZX  O  C;    ZA  O  C. 


Fig.  37 

7.  If  ZXOA  is  taken  from  ZXOB  (Fig.  37),  what 
angle  is  left  ?    Answer  by  an  equation. 

8.  Show  by  an  equation  the  angle  that  is  the  diflference 
between  ZX  O  Y  and  ZXOA  (Fig.  37);  ZA  O  Y  and 
ZAOB;    ZXOY  and    ZB  O  Y. 

■  9.  From  a  point,  O,  on  a  straight  line  X  Y  (Fig.  38)  draw 
three  lines  as  in  the  figure.  Find  the  number  of  degrees  in 
each  angle,  first,  by  estimating,  then  by  measuring  with  the 
protractor.    Tabulate  results  as  in  Fig.  39,  and  find  the  sum. 


B 


Angle 

Estimated 

Measured 

I 

40° 

39° 

2 

3 

90° 

4 

Sum 

X  0  ] 

Fig.  38 

Fig.  39 

10.  What  should  be  the  sum  of  all  the  angles  about  a 
point,  on  one  side  of  a  straight  line?  Check  the  sum  in 
exercise  9. 

Note. — To  check  results  means  to  show  that  they  are  correct. 
Students  should  habitually  check  results,  because  correct  results  show 
that  the  work  is  probably  correct. 


36 


First-Year  Mathematics 


Fig.  40 


11.  Draw  four  lines  from  a  point  (Fig.  40). 
Find  the  number  of  degrees  in  each  angle,  first, 
by  estimating,  then  by  measuring  with  a  pro- 
li  actor.  Tabulate  results  as  in  exercise  9,  and 
check. 

12.  What  is  the  sum  of  the  angles  that  just 
fill  the  plane  about  a  point  ? 


The  Sum  of  the  Angles  about  a  Point 
I.  Find  the  number  of  degrees  in  each  angle  of  Fig.  41. 


WTiy? 


(1) 


Subtracting  30 
Dividing  by  6 
Whence 


Fig.  41 
We  may  write 

X+2X-\-2X-\-?>-\-X-\-22  =  T,6o 

This  equation  may  be  written 

X+2X-\-2X-\-X-\-i 

Combining  like  terms     6ic  +  30  =  36o. 
6a;  =  330. 

»=   55- 

2a;  =  110. 

2;(;  +  8  =  ii8. 

ji;+22=  77. 

Check:  55  +  110+118  +  77=360. 

2.  With  a  protractor  measure  the  angles  of  Fig.  41  to  see 
if  the  figure  in  the  book  is  accurately  drawn. 

3.  To  obtain  the  second  equation,  problem  i,  from  the 
first  equation,  the  order  of  the  terms  was  changed  on  the  left 
side  of  the  first  equation.     Why  may  this  be  done  ? 


-22  =  360.     Why? 

(2) 

(3) 

What  axiom  is  used  here  ? 

(4) 

What  axiom  ? 

(5) 

Why? 

(6) 

Why? 

(7) 

Why? 

(8) 

The  Equation  Applied  to  Angles 


37 


4.  Perform  the  following  additions  and  subtractions: 

(i)  8  +  7+3+6  (5)  8  +  7-3+6  (9)  a  +  7 

(2)  8+3+6  +  7  (6)  8  +  7+6-3  (10)  7+a 

(3)  7+6+8+3  (7)  7-3+8+6  (II)  3a  +  7+a 

(4)  6+8+3+7  (8)  6-3+8  +  7  (12)  3a+a  +  7. 

5.  In  problem  4  compare  the  results  of  (i),  (2),  (3),  and 
(4);  of  (5),  (6),  (7),  and  (8);  of  (9)  and  (10);  of  (11)  and  (12). 

6.  In  the  following  problems  perform  the  additions  and 
subtractions  in  different  orders.  Compare  the  results  of  each 
problem  for  the  different  orders,  each  term  as  it  changes  posi- 
tion carrying  with  it  the  sign  next  before  it : 

(i)  12+7-4+6  (3)  ^x+s-^x-[-2 

(2)  -jx+s-^x  (4)  7^+5-3^-2. 

7.  What  conclusion  do  you  draw  from  problems  4,  5, 
and  6  ? 

28.  The  order  0}  terms  of  an  expression  may  be  changed, 
without  changing  the  value  of  the  expression,  provided  each 
term  as  it  changes  position  carries  with  it  the  sign  next  before  it. 

I.  Find  the  number  of  degrees  in 
each  angle  of  Fig.  42,  writing  the  solu- 
tion as  in  the  foregoing  problem  i. 

All  the  angular  space  about  a  point 
in  a  plane  is  divided  into  angles  repre- 
sented by  the  following  expressions;  find 
X  and  each  angle  in  degrees.  With  pro- 
tractor draw  figures  for  2,  3,  and  4: 

2.  2X,  X,  ^x-\-^o,   180— 331; 
X,  36+531;,   3:»£;-9 
7,x,  sx,   5^+45,   ^1-x 
i<,x^-i6\,  3']^—2x,   8:x;— 9 
T,x,   5:)c  +  26f,   2X,   9X+143I 
7X+24,    14^+531,    i2of-3:x;. 


Fig.  42 


38  First-Year  Mathematics 

All  the  angular  space  about  a  point  in  a  plane,  on  one  side 
of  a  straight  line,  is  divided  into  angles  represented  by  the 
following  expressions.  Find  x  and  each  angle  in  degrees. 
Draw  figures  for  8,  9,  and  10: 

8.  X,    ^x,    'JX—2 

9.  9^,  X,   37— 2JC,   5:^1;— 26 

10.  Sjc,  48— 3:v,  5ic— 22,  431;— 14 

11.  25§+5:!c,  8:x;+8f,  tx,  ()\—2x 

12.  2>^,   2(:!C+9),  X,  \2—x 

13.  2x^   2(:x;+io),  x—1%,   z{z^—x) 

14.  Z{x-Z),  x-\rZl,   2(41 -x) 

15.  2.8;x;4-39.33,   i.2jf— 32.09,  x+7.16 

16.  6.93:x:,   4.82:x:,    1.27^  +  5.09,    138.91 —9. 02.r, 

Exercise  V 
Solve  the  following,  doing  all  you  can  mentally: 

;•  3^+5-^=^5  ,,,  :+8-^=io 

\/2.    ^X  —  t-\rX  =  2\  3  5 

V  3.  8-:x;+3X=32  g^    5^ 


>/4.  8/  — 10— 2^=50 


13.  ^-^=28 


v5-  i5-6/4-8/=2S  3. 

•6.  6/-7+4/  =  i3  M-  -^ — 15  +  7  =  52 

V^.  iy-Zy^rXoy=i^  3.7/-3.6-2.9/  =  2.4 

8.  2(x-3)  +  i2  =  i8  7  3-7^    3-0     2.9^       .4 

9.  8+5(5  +  7)=63  '^'  2^+4(^+io)+3^=i3o 

l/io.  i8r  +  i3-ior  =  75  17.  3f^_5.^+^^8 

^  4      12     3 

II-  ~  '  S~9  18.  7arx;— a:c+a  =  7a. 

Adjacent  Angles 

29.  The  point  at  which  the  sides  of  an  angle  meet  is  called 
the  vertex  of  the  angle. 

Vertex  is  a  Latin  word  meaning  "turning-point." 


The  Equation  Applied  to  Angles 


39 


I,  On  tracing  paper  make  a  trace  of  Zx  (Fig.  43),  and 
fit  this  trace  on  Z.y.  How  do  the  angles  compare  in  size? 
Does  the  size  of  an  angle  depend  on  the  lengths  of  its  sides  ? 
Measure  Z.x  and  Ay  with  a  protractor. 


Fig.  43 

2.  Read  the  vertex  and  the  sides  of  ZA  O  B  (Fig.  44);  of 
ZBOC. 

3.  Do  the  angles  of  Fig.  44  have  a  side  in  common?  A 
common  vertex  ? 

30.  Two  angles  that  have  the  same  vertex  and  a  common 
side  between  them  are  adjacent  angles. 

The  sides  which  are  not  common  are  called  the  exterior 
sides. 

1.  Are  Z&  and  Zc  (Fig.  45)  adjacent?  Zc  and  Ad} 
Z6  and  Aa}     Z6  and  /.d} 

2.  Read  the  exterior  sides  of  Z^i;  and  Ay  (Fig  44);  the 
common  side. 


Fig.  46 


Fig.  45 
31.  The  angles  x  and  y  (Fig.  46)  may  be  added  without 
measuring  them,  by  placing  them  adjacent.  , 


40 


First- Year  Mathematics 


I.  Express  by  an  equation  that  ZA  O  C  (Fig.  47)  equals 

the  sum  of  two  angles. 


O  A 

Fig.  47  Fig.  48 

2.  If  the  angle  y  (Fig.  47)  is  turned  over  O  B  as  a  hinge, 
so  as  to  make  O  C  lie  along  the  dotted  line  O  C  (Fig.  48), 
express  the  value  of  angle  A  O  C  in  terms  oi  /^x  and   Z.y. 

3.  Draw  two  angles,  a  and  b,  and  show  how  to  find  the 
sum  without  measuring  them.  Show  how  to  find  the  differ- 
ence without  measuring. 

4.  Show  the  sum  and  the  difference  of  angles  by  folding 
or  cutting  paper. 

32.  With  a  protractor  draw  an  angle  of  60°. 

Examine  Figures  49  and  50  and  construct  on  a  line,  A  B,  an  angle 
of  60°,  like  B  O  C,  Fig.  51. 


Fig.  49 


t—h 


The  Equation  Applied  to  Angles  41 

1.  With  a  protractor  draw  the  following  angles,  marking 
on  each  angle  the  num- 
ber  of   degrees,  as    in  /^ 
Fig.5i:8o°;92i°;i7o°. 

2.  With  a  protractor 
draw  adjacent  angles  of 
75°  and  85°;   of   io3i° 

and  57i°;   of  31  J°  and        

2^°.     Check  the  work,      ^  ^  .  ■» 

first,  by  finding  the  sum  ^'^*^*  5^ 

arithmetically,  then,  by  measuring  the  sum  with  a  protractor. 

3.  Draw  adjacent  angles  of  56°  and  124°;  of  ig^°  and 
i6oJ°;  of  92°  and  88°.  With  a  ruler,  or  straight  edge,  see  if 
the  exterior  sides  of  each  pair  of  angles  form  a  straight  Une. 
What  term  is  applied  to  each  angle-sum  ? 

4.  Draw  two  straight  lines  cutting  each  other  so  as  to 
make  a  pair  of  adjacent  angles  equal  (with  the  aid  of  a  pro- 
tractor), and  show  that  the  angles  are  right  angles. 

33.  If  two  straight  lines  intersect  (cut  each  other)  making 
a  pair  of  adjacent  angles  eqtml,  the  angles  are  right  angles,  and 
either  line  is  perpendicular  to  the  other. 

34.  An  angle  less  than  a  right  angle  is  called  an  acute 
(sharp)  angle.  An  angle  greater  than  a  right  angle  and  less 
than  two  right  angles  is  called  an  obtuse  (blunt)  angle. 

1.  Draw  two  adjacent  acute  angles  whose  sum  is  a  right 
angle.     Point  out  two  perpendicular  lines. 

2.  Draw  two  adjacent  obtuse  angles  whose  sum  is  3  right 
angles. 

3.  Draw  two  adjacent  angles,  one  obtuse  and  the  other 
acute,  whose  sum  is  two  right  angles. 


43 


First-Year  Mathematics 


4.  Draw  a  triangle  of  which  all  angles  are  acute.  From 
the  vertex  of  each  angle  draw  a  line  at  right  angles  to  the 
opposite  side. 

5.  Draw  a  triangle  having  an  obtuse  angle,  and  draw  per- 
pendiculars as  in  problem  4. 

6.  Draw  a  triangle  having  a  right  angle,  and  draw  per- 
pendiculars as  in  problem  4. 

7.  How  many  jiltitudes  does  a  triangle  have  ?  How  many 
bases  ? 


Supplementary  Angles 

'  35.  Two  angles  whose  sum  is  a  straight  angle  (180°)  are 
supplementary  angles.  Either  angle  is  said  to  be  the  supple- 
ment of  the  other. 

If  the  supplementary  angles  are  adjacent  they  are  called 
supplementary  adjacent  angles. 

1.  In  Fig.  51  read  the  angle  which  is  the  supplement  of 
ZBOC;  of  ZCOA. 

2.  Draw  two  supplementary  adjacent  angles. 

3.  On  tracing  paper  make 
a  trace  of  Z^  and  oi  Z.y  (Fig. 
52)  so  that  the  angles  on  the 
tracing  paper  are  adjacent. 
Show  with  a  ruler,  or  straight 
edge,  whether  the  angles  are 
supplementary.  In  the  same 
way  show  whether  Z.a  and 
Zb  are  supplementary;  Ac 
and  Ad. 

'  ^^  4.  Are  50°  and  130°  sup- 

plementary?   37°  and  133°?    60°  and  120°?    90°  and  90°? 


The  Equation  Applied  to  Angles  43 

5.  How  many  degrees  are  there  in  the  supplement  of  an 
angle  of  45°?    Of  120^°?    Of  90°?    Of  a°?    Of  i8o°-;c°? 

6.  Write  the  supplement  of  an  angle  of  x  degrees. 

7.  Write  the  supplement  of  a°;   of  b°;   of  ^dP;   of  --;   of 
y°+z°. 

8.  If  angles  of  120°  and  a°  are  supplementary,  what  does 
a  represent  ? 

9.  If  :c°+8o°  =  i8o°,  what  is  the  supplement  of  x°  ?    What 
is  the  value  of  x°  ? 

10.  In  the  equation,  a°+6°  =  i8o°,  what  is  the  supplement 
ofa°?    Of6°?    Why? 

11.  State  by  an  equation  that  the  following  pairs  of  angles 
are  supplementary: 

(i)  x°  and  60°  (4)  50°  and  x°  +  'jo° 

(2)  70°  and  y°  (5)  2X°+f  and  27^°-2° 

(3)  6°  and  c°  (6)  fx°  and  Vx°  +  ii2i°. 

12.  x-\-^  degrees  is  the  supplement  of  2:x;+27  degrees. 
Find  X,  x-\-^,  and  2^+27. 

We  may  write  x  +  ^  +  2X  + 2^  =  180.  Why?  (i) 

Combining  like  terms  3:11;  +  30  =  180.  Why?  (2) 

Subtracting  30  3»  =  150.  What  axiom  is  used  ?  (3) 

Dividing  by  3  x=  50.  What  axiom?  (4) 

Whence  «+3=   53-  Why?  (5) 

2X+2J  =  12'J.  (6) 

Check:  53  +  127  =  180. 

13.  ic°  is  the  supplement  of  x°  +84°.    Find  the  angles. 

14.  One  of  two  supplementary  angles  is  98°  larger  than 
the  other.     Find  the  angles. 

15.  One  of  two  supplementary  angles  is  27°  smaller  than 
the  other.     Find  the  angles. 


44  First- Year  Mathematics 

i6.  The  difference  of  two  supplementary  angles  is  iio°. 
Find  them.  [Suggestion:  Let  oif^  be  one  angle  and  5f°  +  iio° 
the  other.] 

17.  Find  two  supplementary  angles  whose  difference  is 

21°;  36^;  nk"";  ^. 

18.  The  difference  between  an  angle  and  its  supplement 
is  37°.     Find  the  angle. 

19.  How  many  degrees  are  there  in  the  angle,  jc°,  if  it  is 
the  supplement  of  sx°  ?    Of  ']x°  ?    Of  3^^°  ? 

20.  How  many  degrees  are  there  in  an  angle  that  is  the 
supplement  of  4  times  itself  ?  Of  8  times  itself  ?  Of  10  times 
itself?     Of  2^  times  itself?     Of  f  of  itself?     Of  \  of  itself? 

21.  Write  in  algebraic, language — 

the  double  of  an  angle,  x 

15°  added  to  3  times  the  angle 

29°  subtracted  from  6  times  the  angle 

4  times  the  sum  of  the  angle  and  13° 

two-thirds  of  the  sum  of  the  angle  and  1 7°. 

22.  Write  in  algebraic  symbols — 

(i)  the  supplement  of  an  angle,  x 
(2)  5  times  the  supplement 
{2,)  2)  times  the  supplement 

(4)  14°  added  to  3  times  the  supplement 

(5)  16°  subtracted  from  3  times  the  supplement 

(6)  the  supplement  increased  by  10° 

(7)  the  supplement  diminished  by  18° 

(8)  the  supplement  divided  by  4 

(9)  one-third  of  the  supplement 

(10)  17°  added  to  the  supplement 

(11)  20°  added  to  one-third  of  the  supplement 

(12)  19°  subtracted  from  f  of  the  supplement. 


4 


The  Eqtiation  Applied  to  Angles  45 


23.  If  an  angle  is  doubled  and  its  supplement  is  increased 
by  20°,  the  sum  of  the  angles  obtained  is  280°.  Find  the  two 
supplementary  angles.  [Suggestion:  Let  x  be  one  angle,  and 
180— X  the  other;  then  by  the  conditions  of  the  problem 
2:v  +  i8o— ;x:+20  =  28o.] 

y  ,'24.  If  an  angle  is  trebled,  and  its  supplement  is  diminished 
by  112°,  the  sum  of  the  angles  obtained  is  168°.  Find  the 
supplementary  angles. 

\/25.  The  sum  of  an  angle  and  ^  of  its  supplement  is  90°. 
Find  the  angle. 

^6.  If  an  angle  is  increased  by  12°,  and  its  supplement  is 
divided  by  5,  the  sum  of  the  angles  obtained  is  80°.  Find  the 
supplementary  angles. 

^27.  If  20°  is  added  to  5  times  an  angle,  and  15°  is  sub- 
tracted from  2  times  the  supplement  of  the  angle,  the  sum  of 
the  angles  obtained  is  401°.     Find  the  supplementary  angles. 

.  Find  X  and  each  angle  of  the  following  pairs  of  supple- 
mentary angles.  With  protractor  draw  figures  for  28,  29, 
and  30: 

28.  4x,   88 -|- Ire  2X  AX 

\\         J      1  35-  77+—,   59+7- 

29.  ix+\^,  ?>2-^x  3  5 

30.  f:x:-h29,   gT-\x  .    2x  ,         x      ^ 

36.  —+93,   --18 
X  .     ,      X  3  2 

31-  -  +  161,   --26 

32  -IX  ^       2X 

X  X  37-  23+—,   136-— 

32.  -+97»   7+69  ^ 

3  4  38.  3(^-f8),    K4^+88) 

Zi-  ^+37,    109+V  39-  ^(2^+15),   78+f^ 


4  3 


40.  hix-z6),   J(279-|-2:k;) 


XX 

^^'     ^^5'    3"^  ^  41.  i(3^+i48),   ^2^+327). 


46 


First- Year  Mathematics 
Exercise  VI 


Solve  the  following: 

1.  ^+88+-=i8o 

5  5 

4'  ,  3' 

2.  — +  IO  — — =i(; 

7  14 

3-  -  +  i6+--i4  =  7 

4.  --15+^=8 
3  4 

5-  9H i2+x=9 

6.  2{x-s)  +  \{x->ri)=2> 
7-  5(^-5)  +  (^+3)  =  2 

8.    -^ 20  +  (z  +  I2)=Q 

2  \  /        7 

0. 1-8  =  12 


ZX-2 
lO.    ^ I=f 


„.  5fch4)+8^,3 


12.  9:x; 


-r-i  =  7i- 


7  14 

^ — it 

14. |-io+^  =  iof 

2 

3  4 

,7.  7jZ±5)_3ZZi_9=4. 


Vertical  Angles 

36.  Two  angles  having  a  common  vertex,  and  having  sides 
j5     in  the  same  straight  line,  but  in  opposite  direc- 
tions, are  called  opposite  or  vertical  angles  (as 
X  and  2,  Fig.  53). 

1.  Read  both  pairs  of  vertical  angles    in 
Fig.  53- 

2.  On  tracing  paper  make  a  trace  of  angles 
'y  and  z  (Fig.  53).     Put  this  trace  on  angles  x 

and  w  and  see  whether  Z.z  coincides  with  (fits 
on)  /Lx,  and  Z.y  with  Aw.    How  do  the  ver- 
tical angles  compare  in  size? 


The  Equation  Applied  to  Angles  47 

3.  Test  your  conclusion  in  problem  2  by  drawing  two 
intersecting  straight  lines  and  measuring  both  pairs  of  vertical 
angles  with  a  protractor. 

4.  Are  Z^  X  and  y  (Fig.  53)  adjacent?  Are  they  supple- 
mentary ?    What  is  their  sum  ? 

5.  Are  Z^  y  and  z  (Fig.  53)  supplementary  adjacent  ? 
What  is  their  sum  ? 

6.  From  problems  4  and  5  show  that 

x+y=y-i-z. 

7.  Show  how  to  get  the  equation 

x=z 
from  the  equation  in  problem  6.     What  axiom  must  be  used  ? 

8.  Using  the  equation 

y+x=x+w, 
show  as  in  problems  4,  5,  6,  and  7  that 

y=w. 

37.  The  problems  of  section  36  show  the  truth  of  the 
following  theorem  or  proposition: 

Theorem.  //  two  lines  intersect,  the  vertical  angles  are 
equal. 

1.  Show  that  the  above  theorem  is  true  for  any  two  inter- 
secting straight  lines,  no  matter  what  angles  they  make. 

2.  State  the  three  ways  in  which 
the  truth  of  the  foregoing  theorem 
was  shown  in  §36.  By  which  of  these 
ways  was  the  truth  of  the  theorem 
shown  for  any  pair  of  vertical  angles  ? 

3.  Find  the  values  of  the  four  angles 
of  Fig.  54.  Fig.  54 


48 


First-Year  Mathematics 


We  may  write 

7f«  +  3i^  +  92-3f«  =  i8o.     Why? 

Equation  (i)  may  be  written 

7§«  +  3i«-3t«+92  =  i8o. 

7§x  +  Q2  =  i8o. 

Subtracting  92  7J^=  88. 

Multiplying  by  3  22a;  =  264. 

Dividing  by  22  x=   12. 

^lx  +  iifc  =  ^l^  12  +  3J.  12  =  02  +  42  =  134. 
92-3|»=        92-3!  •  12  =  92-46=  46. 

The  other  two  angles  are  also  134°  and  46°.     Why? 
Check:  134+46  +  134  +  46  =  360. 


(I) 


Why? 

(2) 

Why? 

(3) 

What  axiom  ? 

(4) 

What  axiom  ? 

(5) 

What  axiom  ? 

(6) 

Fig.  55 

3^+37  and  5^+7. 

Since  the  given  angles  are  vertical  angles, 


4,  Find  the  four  angles  made  by 
two  intersecting  straight  lines,  if  two 
adjacent  angles  are  9X+41  and 
5:x;— 29.  Check  results  by  drawing  a 
figure. 

5.  Find  X  and  the  four  angles 
made  by  two  intersecting  straight 
lines,  if  two  vertical  angles  (Fig.  55)  are 


5^  +  7=3^  +  37- 

(I) 

Subtracting  7 

5x  =  3JC  +  30.     What  axiom  ? 

(2) 

Subtracting  3* 

2«  =  30.             What  axiom  ? 

(3) 

Dividing  by  2 

ic  =  15.             What  axiom  ? 

SX+  7=5-15+   7=82. 
3«+37=3- 15  +  37=82. 

(4) 

Each  of  the  other  two  angles  is  180-82  =  98.     Why  ? 
Check:  82  +  98  +  82  +  98  =  360. 

Find  X  and  each  of  the  following  angles  made  by  two 
intersecting  straight  lines.  Draw  figures  for  6,  7,  16, 
and  17: 


The  Equation  Applied  to  Angles  49 

VERTICAL  PAIRS 

6.  'jx+2']  and  4^+87 

7.  7)X—i']  and  :x;  +  io3 

8.  fx+i6|  and  |rv+24| 
■9.  2y\x-i3  and  It^t^+57 

10.  ^x+f:r  and  f:x;+42 

11.  fjc+^x— 28  and  X 

%x  J  5^  , 

12.  5:x;+^^  and  ^^ — ^130 

4  2 

x-i. and h  66 

-^46  3 

14.  -+7-  and  -+  18 

36  4 

,5.7^_3^  and^+8f. 

^24  5 

ADJACENT  PAIRS 

^    X    X  ^        1  J  3^    ^ 

16.  -+-+172^      and 

52  10    4 

X  J  3^  , 

17.  X —  and  - — 1-90 

'  7  4 

18.  — +-  and  iS4i-T 

5      3  o 

10.  — VAX  and  87 

^4  3 

20.  6|:x:— 2i:x;  and  4f.:>f— 365 

21.  — \-2X  and  -7+35 
4  6 

22.  yV(26^-i43)  and  A(i2:x;+33) 
,3.  ,g+|)  and  i(309-^) 

24.  2( — \-x\  and  ■J^(:x;  +  2io) 

25.^(261-53;)       and  4(3:+^). 


50  First-Year  Mathematics 

Exercise  VII 
Solve  the  following,  doing  all  you  can  mentally. 

•^  ^  14.  8z  — =2z  +  i7 

2.  5^:— 6=2:x;  3 

3.  4/+3=/  +  i2  5^_J__^  ,  1 

4.  35  —  2=254-7  8      16     2 
5-  7>'-7=3:V+2i                 ^^    J^+i^^ 

6.  5(^-3)  =3^+3  '     3 

7.  2(2X  +  i)=3X+5  17.  4(:x:-7)=3:x;  +  i4 

8.  6(2-7)  =Z+8  n      ^       ^ 

18. =  iT-x 

9-  7(32-2)  =  52  +  2  3     5 

10.  io(r-i)  =  ior-5  7(^+3)  ,  ,     s 

*  /     .     \     0/  \  ^9- 1-5=-+28 

*ii.  s(x+i)=8ix-4)  ^12  3 

?:x;  20.  a5(;+o=9a 

12.   - — 1-4=^  +  2  I  U        i. 

5  21.  62-36=56 

4^ ,  ,  h ,  r    1^ 

i-i. }-io=3(;+4  22,  —  +  0=—. 

7  3  5 

♦Suggestion. — After  multiplying  as  indicated,  subtract  2,x  from  both 
sides  of  the  equation,  and  add  32. 

It  is  sometimes  convenient,  in  solving  equations,  to  add  or  to 
subtract  such  a  number  or  such  numbers  that  the  term  containing  the 
number  whose  value  is  required  may  stand  alone  on  the  right  side. 

Complementary  Angles 

38.  Two  angles  whose  sum  is  a  right  angle  are  comple- 
mentary angles. 

Either  angle  is  said  to  be  the  complement  of  the  other. 

1.  What  is  the  complement  of    Z.a   (Fig. 
56)?    Of  Z6? 

2.  Draw  two  adjacent  complementary   an- 
gles.    May  either  angle  be  obtuse  ?    Point  out 

Fig.  56        two  perpendicular  lines. 


The  Equation  Applied  to  Angles 


SI 


3.  On  tracing  paper  make  a  trace  of  Z.x  and  of  /Ly 
(Fig.  57)  so  that  the  angles 
on  the  tracing  paper  are 
adjacent.  With  a  protractor 
show  whether  the  angles  are 
complementary.  In  the  same 
way  test  whether  Za  and  Ah 
are  complementary;  Z.c  and 
Ad. 

4.  Show  whether  22°  and 
68°  are  complementary;  43 f° 
and  46§°;  89!°  and  f. 

5.  What  is  the  comple- 
ment of  60°?  Of  30°?  Of 
lof  ?  Of4Sf°?  Ofa°?  Of 
go°-x°}  Fig.  57 

6.  Write  the  complement  of  n  degrees. 

7.  Write  the  complement  of  t?';  of  3C°;  of  ^— ;  oix°-\-y°; 

of^ ;  of  7 (a +&)  degrees;  of  53;^  degrees;  oi 'jy^  degrees; 

of  ^x^—sy*  degrees. 

8.  If  angles  of  40°  and  dF*  are  complementary,  how  many 
degrees  does  d  stand  for  ? 

9.  If  y°  +  'jo°=go°,  what  is  the  complement  of  y°  ?    Why  ? 
What  is  the  value  of  y°  ? 

10.  In  the  equation,  c°+dP—go°,  what  is  the  complement 
ofc°?    Oid9?    Why? 

11.  State  by  equations  that  the  following  pairs  of  angles 
are  complementary: 


52 


First-Year  Mathematics 

(i)  y°  and  50°  (2)  30°  and  2°  (3)  iiP  and  x° 

(4)  a°+3o°  and  o°-2o°  (5)  2x°  +  f  and  5^°-2° 

(6)  3(:x;+7)  degrees  and  5(2:^;— 8)  degrees 

(7)  §^""157  degrees  and  26^:^+431  degrees. 

ft  ft        ft  fl 

12.  If  — h5S°  is  the  complement  of ,  find  »,  — f-55°, 


,  n    n 
and 


3     4 
We  may  write 


3     4 
Rearranging  the  terms  of  equation  (i) 


n  n    n 

-  +  55+---=  90-    Why?  (1) 


n     n    n 

-+ +  55=  90.     Why?  (2) 


234 
Combining  the  M-terms 

^  +  55=  90-  (3) 

Subtracting  55  T^^  35-     Why?  (4) 

Multiplying  by  12  7w=420.     Why?  (5) 

Dividing  by  7  »=  60.     Why?  (6) 

«  .  60  ,  „ 

-  +  55=  — +  55=85- 
2  2 

w     n     60    60 

3     4~  3       4 
Check:  85  +  5  =  90. 

13.  x°  is  the  complement  of  .-v;°+48°.     Find  the  angles. 

14.  One  of  two  complementary  angles  is  24°  larger  than 
the  other.     Find  the  angles. 

15.  One  of  two  complementary  angles  is  28°  smaller  than 
the  other.    Find  the  angles. 

16.  The  difference  of  two  complementary  angles  is  83°. 
Find  them. 

17.  Find  two  complementary  angles  whose  difference  is 

21°;  36^°;  73i°;  ^. 


The  Equation  Applied  to  Angles  53 

18.  The  "difference  between  an  angle  and  its  complement 
is  27°.     Find  the  angle. 

19.  How  many  degrees  are  there  in  the  angle,  x,  which  is 
the  complement  of  4:^  ?    Of  6:x;  ?    Of  5^:^  ? 

20.  How  many  degrees  are  there  in  an  angle  that  is  the 
complement  of  3  times  itself  ?  Of  7  times  itself  ?  Of  6  times 
itself?    Of  3 i  times  itself ?    Of  |  of  itself?    Of  ^  of  itself ? 

21.  Write  in  symbols — 

the  double  of  the  angle  y 

17°  added  to  3  times  the  angle 

31°  subtracted  from  5'  times  the  angle 

6  times  the  sum  of  the  angle  and  19° 

f  of  the  sum  of  the  angle  and  22° 

5  times  the  angle  minus  16°. 

22.  Write  in  symbols — 

(i)  the  complement  of  an  angle  y 

(2)  6  times  the  complement 

(3)  4  times  the  complement 

(4)  19°  added  to  4  times  the  complement 

(5)  17°  subtracted  from  4  times  the  complement 

(6)  the  complement  of  y  increased  by  12° 

(7)  the  complement  diminished  by  25° 

(8)  the  complement  divided  by  7 

(9)  one-fifth  of  the  complement 

(10)  12°  added  to  the  complement 

(11)  13°  added  to  one-fourth  of  the  complement 

(12)  18°  subtracted  from  f  of  the  complement. 

23.  If  an  angle  is  doubled,  and  its  complement  is  increased 
by  40°,  the  sum  of  the  angles  obtained  is  i6o°.  Find  the 
complementary  angles. 

24.  If  an  angle  is  trebled,  and  its  complement  is  diminished 
by  40°,  the  sum  of  the  angles  obtained  is  130°.  Find  the 
complementary  angles. 


54  First-Year  Mathematics 

25.  The  sum  of  an  angle  and  i  of  the  complement  is  75°. 
Find  the  angle.  ^^ 

26.  If  an  angle  is  increased  by  15°,  and  the  complement  is 
divided  by  3,  the  sum  of  the  angles  obtained  is  75°.  Find  the 
complementary  angles. 

27.  If  20°  is  added  to  3  times  an  angle,  y,  and  6°  is  sub- 
tracted from  f  of  the  complement  of  y,  the  sum  of  the  angles 
obtained  is  102°.     Find  the  complementary  angles. 

The  Sum  of  the  Angles  of  a  Triangle 

I.  Measure  the  angles  of  a  triangle  with  a  protractor  and 
find  the  sum. 

2.  Draw  and  cut  out  a  triangle. 
Tear  off  the  comers,  and  place 
them  as  in  Fig.  58.  What  seems  to 
be  the  sum  of  the  three  angles  of 
the  triangle  ?  Test  with  a  ruler, 
or  straight  edge. 

3.  Draw  tw'o  triangles  different 
in  size  and  shape  from  the  triangle 

X  */^    \  used  in  problem    2,  and  find  the 

sum    of     the   angles   as   in  prob- 

FIG.  58  ,  b  t^ 

lem  2. 

4.  Draw  a  triangle.  From  a  vertex  draw  a  line  (B  O, 
Fig.  59)  at  right  angles  to  the  opposite  side.  Cut  out  the 
triangle  and  fold  so  that  the  vertices 

come  together  at  the  foot  of  the  per- 
pendicular   (at    O,    Fig.    59).     What  y 
seems    to    be    the  sum  of  the    three  /^ 
angles  of  the  triangle  ?  /  [iiV; 

.  -n     ^   .  •    ,  ^-ff    .  •   LJkm^:h 

5.  Draw  two  tnangles  different  in  ^  0 
size    and    shape    from    the    triangle               Fig.  59 


The  Equation  Applied  to  Angles 


55 


used  in  problem  4,  and  find  the  sum  of  the  angles  as  in 
problem  4. 

6.  Draw  a  triangle.  Place  a  pencil  or  stick  in  the  position 
I  (Fig.  60),  noting  the  direction 
it  is  pointing.  Rotate  the  pen- 
cil through  the  Z^  x,  y,  and  z, 
successively,  as  indicated  in  the 
figure. 

Through  what  part  of  a  com- 
plete turn  has  the  pencil  rotated  ? 
Through  how  many  right  angles  ? 
How  many  degrees  ?  _  Fig.  60 

7.  State  by  an  equation  the  number  of  degrees  in  the  sum 
of  Z^  X,  y,  and  z  (Fig.  60). 

39.  If  one  side  of  the  triangle  ABC  (Fig.  61) is  prolonged 
at  each  vertex,  the  angles  w,  s,  and  /  are  called  the  exterior 
(outside)  angles  of  the  triangle  ABC. 

/  Angles  X,  y,  and  z  are  called 

the  interior  (inside)  angles  of  the 
triangle  ABC. 

40.  If  one  angle  of  a  triangle 
is  a  right  angle,  the  triangle  is 
called  a  right  triangle. 

I.  State  by  an  equation  the 
/  sum  of   the   interior   angles   of 

P^^  g^  triangle  ABC  (Fig.  61). 

2.  How  many  degrees  in  x+w  (Fig.  61)?    In  y-^-sl    In 

z+/? 

3.  Show  that 

:x;+w+)'+5  +  z+f  =  i8o  +  i8o  +  i8o=i;4o. 


56 


First-Year  Mathematics 


4.  From  the  equation  in  problem  3  show  that 

w-\-s+t=^6o. 
The  equation  in  problem  3  may  be  written 

x+y+z  +  v}+s+t  =  $<\o.     Why?  (i) 

But  «+y+z  =  i8o.     Why?  (2) 

If  the  left  side  of  equation  (2)  is  subtracted  from  the  left  side  of  equa- 
tion (i),  and  the  right  side  of  equation  (2)  from  the  right  side  of  equation 
(i),  the  result  is 

w+j+<  =  36o.     Why?  (3) 

5.  Translate  equation  (3)  of  problem  4  into  words. 

41.  We  have  considered  the  following  theorems  (proposi- 
tions) about  the  angles  of  triangles: 

Theorem  I,  The  sum  of  the  interior  angles  0}  a  triangle 
is  180°. 

Theorem  II.  The  sum  of  the  exterior  angles  of  a  triangle, 
taking  one  at  each  vertex,  is  j6o°. 

I  I.  Prove     Theorem 

II,  §41,  by  rotating  a 
pencU  as  indicated  in 
Fig.  62. 

2.  The  interior  an- 
gles of  a  triangle  are  -^x, 
X,  dx.  Find  their  values. 

3.  Find  the  value 
of  each  angle  of  a  tri- 
angle, in  degrees,  if  the 
first  angle  is  twice  the 
second,  and  the  third  is 

Pic  52  three  times  the  first. 

4.  Find  the  angles  of  a  triangle  if  the  first  angle  is  6  times 
he  second,  and  the  third  is  \  of  the  first. 


The  Equation  Applied  to  Angles  57 

5.  Solve  a  problem  like  4,  supposing  the  third  angle  to  be 
J  of  the  first;  §  of  the  first;  ^  of  the  first;  f  of  the  first. 

6.  Find  the  angles  of  a  triangle  if  the  first  is  \  of  the  second, 
and  the  third  is  \  of  the  first. 

7.  Find  the  angles  of  a  triangle  if  the  first  angle  is  18° 
more  than  the  second,  and  the  third  is  12°  less  than  the  second. 

8.  The  difference  between  two  angles  of  a  triangle  is  20°, 
and  the  third  angle  is  36°.     Find  the  unknown  angles. 

9.  Find  the  angles  of  a  triangle  if  the  first  is  25°  more  than 
the  second,  and  the  third  is  3  times  the  first. 

10.  Find  the  angles  of  a  triangle  if  the  first  angle  is  double 
the  second,  and  the  third  is  3  times  the  first,  less  9°. 

11.  Find  the  angles  of  a  triangle  if  the  first  is  \  of  the 
second,  and  the  third  is  \  of  the  first,  plus  18°. 

12.  Find  the  angles  of  a  triangle  if  the  first  is  2>h  times  the 
second,  minus  8°,  and  the  third  is  ^  of  the  second. 

13.  Find  the  angles  of  a  triangle  if  the  first  is  6  times  the 
second,  plus  18°,  and  the  third  is  \  of  the  first,  minus  7°. 

14.  If  one  angle  of  a  triangle  is  a  right  angle,  what  is  the 
sum  of  the  other  two  angles  ?     What  are  such  angles  called  ? 

15.  How  many  angles  of  a  triangle  may  be  right  angles  ? 
Obtuse  angles  ?    Acute  angles  ? 

16.  Find  the  values  of  the  acute  angles  of  a  right  triangle 
if  one  angle  is 

(i)  3  times  the  other  (2)  5  times  the  other 

(3)  §  of  the  other  (4)  i^  of  the  other 

(5)  6  more  than  7  times  the  other 

(6)  ^  of  the  other,  diminished  by  ^:^. 

17.  The  acute  angles  of  a  right  triangle  are  -  and  - .    Find 

2         3 

the  values  of  n  and  of  the  acute  angles. 


58 


First-Year  Mathematics 


i8.  The  acute  angles  of  a  right  triangle  are  equal.  Find 
them. 

19.  Find  the  unknown  interior  angles  of  the  triangles  of 
figures  63,  64,  65,  66.  (Suggestion:  Use  the  theorems  on 
vertical  angles,  and  on  the  angle-sum  of  a  triangle.) 

^  ^^ 

3*^ 


20.  The  three  interior  angles  of  a  triangle  are  equal.     Find 
them. 

21.  The  three  exterior  angles  of  a  triangle  are  equal.     Find 
the  value  of  each  exterior  and  interior  angle. 


Exercise  VIII 


Solve  the  following: 

1.  6:x;— 2=26 

2.  ()X  —  %=X 

,3.  sz+6  =  ii 


4.  2(2-3)  =4 
6.  2y-'j=y-{-^ 


The  Equation  Applied  to  Angles  59 

7.  6r-5=4r+7  6a-i 

'  18. |-a=4  — 2a 

8.  -+-=5 

^  19.    2^  +  2=^+4^ 

45     5  ^6 

9. =7 

5      3  _    ,  ,3^     12-7/ 


5     5 

XO.    ---  =  1 


20.  /  +  — =- 

2  2 

21.  6x— a— 2x=3a 


11.  |-5i=8-—  22.  8(:x;-a)=8+4a 

12.  1^  =  1(3-2/)  23.  X \-2X-\-—=^a 

2  2 

'2X  *4~  2  I 

13.  4:x;-i9  =  — - —  +  i  24.  ^-2a+f  =  i-| 


4 


7(5-2^) 


5(;     X 


14.  2:x;-i= _ t-S  ^    a     2a     ^ 

16.  7z-8=6z+-  27.  z+i  =  ^-|+i2 

8  — •?«  „    2  +  i;z 

17.  5a  +  i- — ^+5  28.  ^-+5z=2. 


Summary 

1.  An  angle  is  formed  by  a  line  rotating  in  a  plane  about 
a  point  called  the  vertex. 

An  angle  may  also  be  formed  by  two  intersecting  lines. 

2.  A  straight  angle  is  the  angle  made  by  a  half-turn  of  a 
rotating  line. 

3.  A  right  angle  is  half  of  a  straight  angle,  or  the  angle 
made  by  a  quarter-turn  of  a  rotating  line. 

4.  An  acute  angle  is  less  than  a  right  angle.  An  obtuse 
angle  is  greater  than  a  right  angle  and  less  than  a  straight 
angle. 


6o  First-Year  Mathematics 

5.  A  degree  is  one-nintieth  of  a  rjght  angle. 

6.  Two  angles  are  adjacent  if  they  have  the  same  vertex, 
and  a  common  side  between  them. 

7.  If  two  straight  lines  intersect  so  as  to  make  the  adjacent 
angles  eqtial,  the  angles  are  right  angles,  and  either  line  is 
perpendicular  to  the  other. 

8.  Angles  may  be  added  and  subtracted  without  measuring. 

9.  The  sum  of  the  angles  having  a  common  vertex,  and 
jiist  covering  the  plane  about  the  vertex,  is  360°,  or  four  right 
angles. 

10.  The  sum  of  the  angles  having  a  common  vertex,  and 
just  covering  the  part  of  the  plane  on  one  side  of  a  straight 
line  is  180°  or  two  right  angles. 

11.  Two  angles  are  supplementary  when  their  sum  is  two 
right  angles,  or  180°.  If  the  supplementary  angles  are  adjacent 
they  are  called  supplementary  adjacent  angles. 

12.  The  exterior  sides  of  supplementary  adjacent  angles  lie 
in  the  same  straight  line. 

13-  Opposite  or  vertical  angles  are  angles  that  have  a 
common  vertex,  and  have  sides  in  the  sam^  straight  lines,  but 
in  opposite  directions. 

14.  Two  angles  are  complementary  when  their  sum  is  a 
right  angle,  or  90°. 

15.  The  truth  of  the  following  theorems  has  been  shown: 

Theorem  I.  //  two  straight  lines  intersect,  the  opposite  or 
vertical  angles  are  equal. 

Theorem  II.  The  sum  of  the  interior  angles  of  a  triangle 
is  180°,  or  two  right  angles. 

Theorem  III.  The  sum  of  the  exterior  angles  of  a  triangle, 
taking  one  at  each  vertex,  is  360°,  or  four  right  angles. 


The  EqiMtion  Applied  to  Angles  6i 

1 6.  A  triangle  having  a  right  angle  is  a  right  triangle. 
i^.  At  least  two  angles  of  a  triangle  must  be  acute. 

1 8.  The  acute  angles  of  a  right  triangle  are  complementary. 

19.  The  equation  may  be  used  to  express  the  relations 
between  angles,  and  to  solve  problems  on  angles. 

20.  The  order  of  terms  of  a  polynomial  may  be  changed, 
without  changing  the  value  of  the  polynomial,  provided  each 
term  as  it  changes  position  carries  with  it  the  sign  next  before  it. 


CHAPTER  IV 

POSITIVE  AND  NEGATIVE  NUMBERS 

Uses  of  Positive  and  Negative  Numbers 

42.  The  plus  (+)  and  minus  (— )  signs  are  used  to  denote 
numbers  supposed  to  be  used  in  certain  directions,  called 
directed  numbers. 

1.  The  top  of  the  mercury  column  of  a  thermometer  stands 
at  0°  at  the  beginning  of  an  hour.  The  next  hour  it  rises  $° 
and  the  next  3°.     What  does  the  thermometer  read  ? 

2.  K  the  mercury  stands  at  0°,  and  rises  8°,  then  falls  5°, 
what  does  the  thermometer  read  ? 

3.  Denoting  a  rise  of  10°,  or  of  x°  on  the  thermometer  by 
R  10°,  or  by  R  x°,  and  a  fall  of  10°,  or  of  x°,  by  F  10°,  or  F  x°, 
give  the  readings  of  the  thermometer  after  the  following  changes, 
if  the  top  of  the  column  reads  0°  at  the  start : 

(i)  R    8°  foUowed  by  R    5°         (4)  R  13°  followed  by  F  18° 

(2)  R  12°        "        by  F    9°         (5)  F  .  5°        "        by  R    x° 

(3)  R  16°        "        by  F  12°         (6)  R    a°        "        by  F     h°. 

4.  If  the  change  in  the  mercury  column  is  a  rise,  a  positive 
or  plus  (+)  sign  will  be  written  before  the  number  that  de- 
notes the  amount  of  the  change.  If  the  change  is  a  jail,  a 
negative  or  minus  (  — )  sign  will  be  written  before  the  number. 
If  the  reading  at  the  start  is  0°,  give  the  readings  after  these 
changes: 

(i)  -f  10  followed  by  +  2  (5)  -f:x;  followed  by  +)» 

(2)  +10        "        by  -  2  (6)  +a        "        by  -x 

(3)  +20        "        by  -18  (7)  +fl        "        by  -a 

(4)  +9        "        by  -12  (8)  -a        "        by  -x. 

62 


Positive  and  Negative  Numbers 


63 


5.  On  a  winter  day  the  thermometer  was  read  at  9  A.  m. 
and  every  hour  afterward  until  5  o'clock.  The  hourly  read- 
ings were  -5°,  0°,   +2°,   +8°,   +10°,   +10°,   +5°,  0°,   -5°. 


40* 

jj* 

Jo" 

nr 

20* 

+ 

15* 

+ 

la* 

/ 

^ 

\ 

+ 

«• 

lAM. 

} 

i 

f 

1 

t 

JAM. 

Ir^ 

/ 

11 

a 

im 

N 

i 

f 

/ 

\ 

Fig.  67 

On  squared  paper  the  readings  were  marked  off  from  hour 
to  hour,  calling  one  vertical  space  5°.  The  points  were  con- 
nected as  shown.  How  did  the  mercury  change  from  9  to  10 
o'clock?  From  10  to  11  o'clock?  From  11  to  12?  From 
12  to  I  ?  From  i  to  2  ?  From  2  to  3  ?  From  3  to  4  ?  From 
4  to  5  ?  Tell  for  each  hour  whether  the  change  was  a  rise, 
or  a  fall. 

6.  Draw  a  line  to  show  the  following  hourly  readings,  be- 
ginning at  8  A.  M.  Indicate  the  hourly  changes  in  amount  by 
figures,  and  in  direction  by  the  +  and  —  signs: 

+  2°,   -2°,   -4°,   -2°,   4-2°,    +4°,    +4°,    +4°,    +8°,    +10°. 

7.  The  average  monthly  temperatures  for  a  northern  town  are 
Jan.      -  4°  May  +42°  Sept.  -f48° 

Feb.     -  7°  June  4-52°  Oct.    4-37° 

Mar.    4-14°  July    4-62°  Nov.  -1-25° 

April    4-26°  Aug.  -f  60°  Dec.   4-  2°. 

Using  a  convenient  scale,  draw  the  temperature-line. 


64  First-Year  Mathematics 

43.  Drawing  temperature-lines  is  called  graphing,  or 
plotting. 

1.  The  daily  average  temperatures  for  14  days  at  a  certain 
place  were  +8^  0°,  -10°,  +12°,  -6°,  +14°,  +15°,  +2°, 
—5°,  +15°,  +20°,  0°,  0°,   +10°.     Graph  these  readings  to 

'a'convenient  scale. 

2.  A  bicyclist  starts  from  a  point  and  rides  r8  miles  due 
northward  (+18  mi.)  then  10  mi.  due  southward  (—10  mi.); 
how  far  b  he  then  from  the  starting-point  ? 

3.  State  how  far  and  in  what  direction  from  the  starting- 
point  a  bicyclist  would  be  after  rides  indicated  by  each  of 
these  pairs  of  records: 

(i)  4-10  mi.  then  —  8  mi.  (3)  -F-ioo  mi.  then  -1-50  mi. 

(2)  —20  mi.    "     -f-20  mi.  (4)  +    ami.    "     +  b  nd. 

4.  How  far  and  in  what  direction  from  the  starting-point 
is  a  traveler  who  goes  eastward  (-h)  or  westward  (— )  as  shown 
by  these  pairs  of  numbers: 

(i)  -fi6  mi.  then  —  6  mi.  (4)  -\-a  mi.  then  +c  mi. 

(2)  —18  mi.     "     -I-28  mi.  (5)  -\-m  mi.     "     —n  mi. 

(3)  — m  mi.     "     -H  3  mi.  (6)  — m  mi.     "     +» mi. 

5.  A  car  in  the  middle  of  a  moving  train  is  drawn  forward 
with  a  force  of  8  tons  and  at  the  same  time  it  is  puUed  back- 
ward with  a  force  of  7^  tons.  The  two  forces  together  are 
equal  to  what  single  force  ? 

6.  Denoting  a  forward  pulling  force  by  F  and  a  backward 
by  B,  give  amount  and  direction  of  a  single  force  equal  to 
each  of  these  pairs  of  forces: 

(i)  F  14  oz.  with  B    6  oz.  (3)  F  25  tons  with  B  15  tons 

(2)  F  20  lb.     "    B  12  lb.  (4)  B  25  tons    "    F  40  tons. 

7.  Denoting  forward-pulling  forces  by  the  positive  or  plus 
(-h)  sign  and  back-pulling  forces  by  the  negative  or  minus 


Positive  and  Negative  Numbers  65 

(— )  sign,  give  the  single  force  which  is  equal  to  each  of  these 

pairs  of  forces: 

(i)   +20  and  —12        (4)  —15  and  +  8        (7)   4-a  and  +b 

(2)  +20    "    -20        (5)  -12    "    -12        (8)  +a    "    -b 

(3)  -15    "     -  8        (6)   +  X    ''    -12        (9)   -a    "     -b. 

8.  A  toy  balloon  pulls  upward  with  a  force  of  9  oz.  If  a 
weight  of  6  oz.  is  attached,  will  the  balloon  rise  or  fall  ?  With 
what  force  ? 

9.  Call  upward  forces  positive,  or  plus  (+),  and  down- 
ward forces  negative,  or  minus  (— ).  State  what  single  force 
will  have  the  same  effect  as  these  pairs: 

(i)  +17  lb.  and  —  7  lb.  (4)  —23  lb.  and  +10  lb. 

(2)  +17  lb.    "    -10  lb.  (s)  +  iclb.    "    +  ylh. 

(3)  -231b.    "    -10  lb.  (6)  +iclb.    "    -jclb. 

10.  Denoting  motion  northward  by  the  positive  or  plus  (+) 
sign  and  motion  southward  by  the  negative  or  minus  (— )  sign, 
and  supposing  a  ship  to  start  on  the  equator  and  sail  as  indi- 
cated, tell  the  latitude  of  the  ship  in  both  amount  and  sign 
for  each  pair  of  sailings: 

(i)  +28°  then  +  2°  (5)  -\-xr  then  -10° 

(2)  +  2°    ''     -18°  (6)  -5C°    "     -10° 

(3)  +12°    "     -12°  (7)  -\-x°    "     -  y° 

(4)  -fi2°    "     -24°  (8)  -x°    "     -  y°. 

11.  A  boy  starts  work  with  no  money.  He  earns  50^ 
(+50^)  and  spends  40^  (—40^).  How  much  money  has  he 
then? 

12.  K  a  man's  debts  be  indicated  by  writing  D  before 
their  amount  and  his  possessions  (assets)  by  P  before  their 
amount,  what  is  the  condition  of  a  man's  affairs  if  his  debts 
and  possessions  are  indicated  by  P  $1,200  and  D  $1,000? 
by  P  $73  and  D  $50?  D  $75  and  P  $60?  D  $300  and 
P  $1,000? 


66  First-Year  Mathematics 

13.  If  water  pushes  (buoys)  a  floating  body  upward  with 
a  force  of  18  lb.,  and  the  body's  weight  pulls  it  downward 
with  a  force  of  10  lb.,  the  two  forces  together  equal  what 
single  force  ? 

14.  If  a  man  was  born  40  b.  c.  and  died  45  a.  d.,  how  old 
was  he  when  he  died  ? 

15.  Denoting  a  date  a.  d.  by  +  and  b.  c.  by  — ,  give  the 
length  of  time  between  these  pairs  of  dates : 

(i)  -  5  to  +10     (3)  -  52  to  -50     (5)   -  150  to    +  150 
(2)  +16  to  +86     (4)  —100  to  +50     (6)  +1600  to    +1900. 

16.  Virgil  was  bom  —70  and  died  —19;  how  old  was  he 
at  death  ? 

17.  The  first  Punic  War  lasted  from  —264  to  -241;  how 
long  did  it  last  ? 

18.  Egypt  was  a  Roman  Province  from  —30  to  +616; 
how  many  years  was  this  ? 

19.  Augustus  was  Emperor  of  Rome  from  —50  to  +14; 
how  many  years  was  he  Emperor  ? 

20.  What  will  denote  the  distance  and  direction  from  your 
school  house  to  your  home,  if  the  distance  and  direction  from 
your  home  to  your  school  house  are  denoted  by  +60  rd.  ? 
4-iimi.  ?    +:!crd.  ?    — 8ord.  ?    — ijmi.  ?    —ami.? 

21.  While  a  freight  train  is  moving  at  the  rate  of  10  mi.  an 
hour  toward  the  south  (  +  10  mi.  an  hr.)  a  brakeman  walks 
along  the  top  of  the  cars  toward  the  north  at  the  rate  of  4  mi. 
an  hour  (—4  mi.  an  hr.).  How  fast  and  in  what  direction 
does  the  brakeman  move  over  the  ground  ?  Answer  with  the 
aid  of  the  plus,  or  minus,  sign. 

22.  The  conductor  of  a  passenger  train  walks  from  the 
front  toward  the  rear  of  the  train  at  the  rate  of  3  mi.  an  hour 
while  the  train  is  running  at  the  rate  of  12  mi.  an  hour.     How 


Positive  and  Negative  Numbers  67 

fast  does  tfie  conductor  move  over  the  ground  ?  Answer  with 
the  aid  of  the  (+),  or  (  — ),  sign,  supposing  that  +  means 
toward  the  north  and  first,  that  the  train  is  running  north; 
then  second,  that  the  train  is  running  south. 

23.  What  does  the  sign  (— )  denote  if  the  sign  (+)  de- 
notes: (i)  above?  (2)  forward?  (3)  upward?  (4)  to  the 
right?    (5)  after?    (6)  east?    (7)  north?    (8)  possessions ? 

44.  The  foregoing  problems  show  the  need  for  distinguish- 
ing numbers  of  opposite  nature.  Positive  and  negative  signs 
afford  a  convenient  means  of  making  this  distinction. 

45.  A  number  that  denotes  both  magnitude  and  direction, 
or  both  size  and  quality,  is  called  a  directed  number,  as  16  units 
east,  or  +16,  10  units  west,  or  —10,  ax  units  downward,  or 
—ax.  If  the  direction  or  quality  is  denoted  by  the  sign  (+)  or 
(— ),  the  number  is  called  a  signed  number. 

Either  of  the  two  opposite  directions,  or  qualities,  may  be 
denoted  by  the  plus  (+)  sign,  whereupon  the  opposite  direction, 
or  quahty,  is  denoted  by  the  minus  (— )  sign. 

46.  The  plus  and  minus  signs  are  also  used,  as  in  arith- 
metic, to  denote  addition  and  subtraction. 

47.  It  is  sometimes  desirable  to  indicate  definitely  that  the 
sign  is  to  denote  direction,  or  quality,  when  used  with  a  num- 
ber. This  is  done  by  inclosing  the  number  together  with  the 
sign  of  quality  in  a  parenthesis,  thus,  (-1-6),  (—8),  {+z^), 
{-2y). 

For  example,  the  expression,  6-f  (-I-2),  means  that  -f2  is 
to  be  added  to  6;  the  expression,  7  — (-I-4),  means  that  4-4 
is  to  be  subtracted  from  7. 

Give  the  meaning  of  the  following: 

1.  6-(4-3)  3.   -5  +  (  +  7)  5.    a  +  {-b) 

2.  8  +  (— 4)  4.   —6  — (—9)  6.  m  —  {—x). 


68 


First-Year  Mathematics 


48.  The  positive  sign  (+)  need  not  always  be  written.  It 
is  generally  omitted  from  the  first  number  in  an  expression. 
The  negative  sign  is  never  omitted.  An  expression  like 
■hx—a,  where  the  first  number,  x,  is  positive,  would  com- 
monly be  written  x—a,  and  is  read  '^x  minus  a."  The  posi- 
tive sign,  when  omitted,  is  said  to  be  "understood." 

Graphing  Data 
I.  Using  a  convenient  scale,  and  calling  the  verticals  age- 


lines,  graph  these  average  heights  of  boys  and  girls: 


Age 

i  yr. 

4 
6 

8- 
10 


Boys 

1.6  ft. 

2.6 

30 

3-5 

4.0 


Girls 

1.6  ft. 

2.6 

3-0 

35 

3-9 


Age 

12  yr. 

14 
16 

18 
20 


Boys 
4.8  ft. 

5 


Girls 
4.5  ft. 


At  what  age  do  boys  grow  most  rapidly  ?     Girls  ? 


u 

io 

b 

J 

'        1 

'4 

3 

■«! 

J      / 

'         / 

j' 

/ 

f 

y  _/ 

y   ^^ 

'^ 

J 

fl 

y 

y^ 

y 

> 

, 

y 

^ 

^ 

Fig.  68 

2.  The  populations,  in  millions,  of  the  United  States  for  each 
10  years  beginning  1790,  are  3.9,  5.3,  7.2,  12.9,  17. i,  23.2, 
31.4,  38.6,  50.2,  62.6,  76.3. 


Positive  and  Negative  Numbers 


69 


37-7      33-9      35- 


32-9      35-8      35 


Graph  these  numbers,  drawing  a  smooth,  free-hand  curve 
through  the  points  and  predict  the  population  for  19 10. 

3.  The  standings  of  the  champion  batters  from  1 900-1 907, 
inclusive,  are  here  given  in  percents,  for 

The  National  League 

38.4      38.2       36.7      35.5      34.9 
The  American  League' 

38.7      42.2       37.6      35.5      38.1 

Graph  these  percents  for  each  league  to  a  convenient  scale, 
both  on  the  same  sheet.    Tell  what  the  percent-line  shows. 

4.  The  monthly  average  rainfall  or  snowfall,  in  inches,  at 
a  certain  place  for  30  years  is  as  follows: 

Jan.      2.8  May  3.59 

Feb.      2 . 30  June  3 . 79 

March  2.56  July  3.61 

April    2 .  70  Aug.  2 .  83 

Graph  these  data  and  tell  what  the  connecting  line  shows. 

5.  Graph  these  average  lengths  of  day  from  sunrise  to 
sunset  in  latitude  42°. 

Hr. 


Sept. 

2.91 

Oct. 

2.63 

Nov. 

2.66 

Dec. 

2.71 

Jan.  16 
Feb.  15 
March  16 
April     15 


9-5 
10.5 
II. 9 
133 


May  16 
June  15 
July  16 
Aug.  16 


Hr. 

14-5 
15.0 
14.9 
139 


Sept.  15 
Oct.  16 
Nov.  15 
Dec.  16 


Hr. 
12.5 
II. 2 

9.6 

9.1. 


6.  Using  the  same  sheet,  scale,  and  dates  as  in  problem  5, 
graph  the  average  day's  lengths 
in  latitude  38° :      9.7       10.8 
14.6        13.7        12.5       II. 2 
in  latitude  45°:      9.1       10.4 
15.3        14. 1        12.6       II. I 

What  differences  in  the  change  of  the  day's  length  in  dif- 
ferent latitudes  do  the  three  graphs  show  ? 


12 .0 

IO-5 

II. 9 

9.6 


^3-3 

9-5 

13-5 


14.4       14-9 


14.9       15.6 


^'. 


J° 


First-Year  Mathematics 


7.  A  ship's  latitude  from  week  to  week  was  +42°,  +38°, 
+30^  +20°,  +12°,  +2°,  -1°,  -6°,  -3°,  +12°.  Graph 
these  latitudes  and  tell  when  the  ship  crossed  the  equator. 

Graphing  Precise  Laws 

I.  Rectangles  are  3  inches  wide  and  their  lengths  are,  4" 
5",  6'',  f,  8",  9",  10",  11",  12",  13".     Calculate  the  areas 
of  these  rectangles  and  plot  them  to  a  convenient  scale.     How 
does  the  graph  show  the  area  to  vary  with  the  base  ? 


/ 

/ 

/ 

, 

/ 

/ 

^ 

3* 

/ 

/ 

/ 

^ 

/ 

/ 

/ 

/ 

m 

^ 

/ 

/ 

V 

y 

IS 

/ 

y 

/ 

^ 

5 

' 

1 

t        S        >        4         S        6        7        S       9       10       11        U      U       14 

Fig.  69 


2.  Write  the  equation  for  the  area,  y,  of  a  rectangle  having 
an  altitude  3  and  a  base  x. 

3.  The  base  of  a  rectangle  is  4  and  the  altitude  is  x.     Write 
an  equation  showing  what  the  area,  y,  must  be. 


Positive  and  Negative  Numbers  71 

4.  In  the  equation  of  problem  3,  let  x  have  the  values  in 
the  first  line  just  below,  and  calculate  the  values  of  y. 

x=i,  2,  3,4,  5,  6,  7,  8,  9,  10 
y=4,  etc. 

Graph  these  values  of  x  and  y  and  state  how  the  area,  y, 
varies  with  the  altitude,  if  the  base  is  4,  a  fixed  number. 

5.  The  side  of  a  square  is  x.  Write  an  equation  showing 
how  to  find  its  area,  y. 

6.  In  the  equation  of  problem  5,  suppose  x  to  have  the 
successive  values  of  the  first  line  shown  here,  and  calculate 
the  areas,  y. 

x=i,  2,  3,  4,  5,  6,  7 
y  =  i,  4,  etc. 

7.  Graph  the  values  of  y  (problem  6)  on  the  i,  2,  3,  4, 
etc.,  lines  and  draw,  free-hand,  a  smooth  curve  through  the 
points. 

8.  The  altitude  of  a  triangle  is  4  and  the  base  is  x.  Write 
an  equation  to  show  its  area,  y. 

9.  Plot  the  values  of  y  (problem  8)  for  the  values  of  x=i, 
2,  3,  4,  5,  etc.,  and  tell  how  the  area  of  a  triangle  depends  on 
its  base  if  the  altitude  is  a  fixed  number. 

10.  Write  an  equation  to  show  the  area,  y,  of  a  triangle 
having  a  base  6,  and  an  altitude  x. 

11.  Graph  the  values  of  y  (problem  10)  for  x  =  i,  2,  3,  4, 
5,  6,  etc.,  and  show  how  the  area  of  a  triangle  depends  on  its 
altitude,  if  its  base  is  a  fixed  number. 

49.  We  have  graphed  the  laws  y=3X,  y=4x,  y=x',  and 
y=2x.    Other  equations  may  be  graphed  just  as  these  laws  were. 
Graph  the  following  equations: 

1.  y=x  4.  ;y  =  2;c  — I  7.  y=x'+2 

2.  y=x-\-x  5.  y=2x+i  8.  y  =  sx 

3.  y=x—z  6.  y=2x-\-3  9.  y  =  ^x—6. 


7 2  First-Year  Mathematics 

50.  It  is  not  necessary  to  have  the  whole  equation  to  make 
the  graph.     Only  the  second  member  is  necessary. 

1.  Graph  x-\-^. 

This  means  take  for  x  the  successive  values,  i,  2,  3,  4,  5,  etc.,  cal- 
culate the  values  oix  +  y,  viz.,  4,  5,  6,  7,  8,  etc.,  and  plot  as  before. 

2.  Graph  the  following  expressions: 

(i)  2X  (4)  2x-\-i  (7)  2:x;— I 

(2)  x-\-2  (5)  x+i  (8)  :x;^ 

(3)  3C-2  (6)  ^-i  (9)  4^+2. 

Adding  Positive  and  Negative  Numbers 

1.  Denoting  distances  traveled  northward  by  positive  num- 
bers, and  distances  traveled  southward  by  negative  numbers, 
find  for  each  of  the  following  cases  the  distances  and  the  direc- 
tion of  the  stopping-point  from  the  starting-point.  When  the 
stopping-point  is  north  of  the  starting-point  mark  the  result 
-|-;  when  south,  mark  the  result  — . 

An  automobile  goes: 

(i)  -hi5  mi.,  then  —10  mi.  (6) 

(2)  -1-15  mi.,     "     -14  mi.  (7) 

(3)  -I- 15  mi.,    "     -20  mi.  (8) 

(4)  -1-25  mi.,    "     -35  mi.  (9) 

(5)  —18  mi,,    "     -J-24  mi.  (10) 

2.  In  the  following  problems  the  numbers  indicate  distances 
traveled  northward,  if  negative;  and  southward,  if  positive. 
The  sum  in  all  cases  must  denote  the  distance  and  direction  of 
the  stopping-point  from  the  starting-point.  Write  the  sums 
with  their  proper  signs: 

(I)         (2)         (3)         (4)         (5)         (6)         (7)         (8) 
+  15       -IS       +15       -IS       -^38       -38       -}-38       -38 
+  8       —  8       —  8       -I-  8       -fi9       —19       —19       +19 

(9)       (10)        (II)        (12)        (13) 
-1-4       —  4       -fii       —  4       -I-12 
+26       —26       —26       +26       —12. 


—  12  mi.. 

then 

I  —10  mi. 

—20  mi.. 

11 

+  15  mi, 

-15  mi.. 

n 

-1-22  mi, 

—20  mi., 

« 

-I-21  mi. 

—22  mi., 

<( 

+22  mi. 

Positive  and  Negative  Numbers  73 

3.  Examine  (i),  (2),  (5),  (6),  (10)  of  problem  2  and  make 
a  rule  for  adding  two  numbers  having  like  signs. 

4.  From  (3),  (4),  (8),  (11),  (12),  and  (13)  of  problem  2 
make  a  rule  for  adding  two  numbers  having  unlike 
signs. 

51.  Sums,  with  their  proper  signs,  of  positive  and  negative 
numbers,  are  called  algebraic  sums.  The  sums  and  differ- 
ences of  numbers  regardless  of  sign,  are  called  arithmetical 
sums  and  differences. 

52.  The  algebraic  sum  of  two  numbers  with  like  signs  is 
their  arithmetical  sum,  with  the  common  sign  prefixed. 

53.  The  algebraic  sum  of  two  numbers  with  unlike  signs  is 
their  arithmetical  difference,  with  the  sign  of  the  larger  num- 
ber prefixed. 

I.  In  the  following  problems  the  positive  numbers  indicate 
gains  and  the  negative  numbers  indicate  losses.  The  sums 
indicate  the  net  change  in  the  man's  capital,  and  whether  the 
net  change  is  an  increase  or  a  decrease.  Find  the  sums  and 
tell  their  meaning : 


(I) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

+50 

+35 

-45 

+  75 

-236 

+Sx 

—14a 

+25 

-38 

—20 

+  13 

+  780 

—6x 

—46a 

-18 

+24 

+60 

-86 

-  95 

-4^ 

+  77« 

-  6 

-15 

+  55 

+  8 

+  45 

+^x 

-  5^- 

2.  State  a  way  of  adding  any  number  of  positive  and 
negative  numbers. 

3.  A  force  of  12  lb.  pulling  toward  the  right  (  +  12  lb.) 
together  with  a  force  of  9  lb.  pulling  toward  the  left  give  a 
combined  pull  equal  to  what  force  ? 


74 


First-Year  Mathemitics 


4.  What  single  force  has  the  same  effect  in  pulling  the 
ring  R  as  the  following  pairs  of  forces  acting  together  ? 


1 1  i.i  1 1 1/ 


a 


D 


Fig.  70 


(i)  +12  lb. 

and 

-  8  1b. 

(9)  -16  lb. 

and 

-  8  1b. 

(2)  -12  lb. 

(< 

+  81b. 

(10)   +  3  lb. 

-12  lb. 

(3)  -10  lb. 

(( 

+  iolb. 

(11)   +  ^Ib. 

+  >'lb. 

(4)  -16  lb. 

(( 

+  13  lb. 

(12)   +  X  lb. 

-  ^'Ib. 

(5)  +14  lb. 

(( 

-17  lb. 

(13)    -  X  lb. 

+  ylh. 

(6)  +  9  lb. 

(( 

-20  lb. 

(14)   -  X  lb. 

-  y\h. 

(7)  +11  lb. 

(( 

+  15  lb. 

(is)   +  X  lb. 

-  x\h. 

(8)  -16  lb. 

(t 

+  12  lb. 

(16)     -2Xlb. 

+  rx;lb. 

5.  A  man  draws  a  pail  of  brick,  weighing  60  lb.,  to  a 
house-top  by  pulling  on  a  rope  which  runs  over  a  pulley,  with 
a  force  of  65  lb.  What  single  force  equals  the  sum  of  the  two 
forces  acting  on  the  handle  of  the  pail  ? 

6.  A  balloon  pulls  upward  on  a  stone,  weighing  6  oz., 
with  a  force  of  8  oz.     What  is  the  sum  of  the  forces  ? 

7.  A  piece  of  iron  weighing  18  lb.,  when  placed  under 
water,  is  pushed  (buoyed)  upward  with  a  force  of  2^  pounds. 
What  is  the  sum  (combined  effect)  of  the  two  forces  together  ? 

8.  An  elevator  starts  at  a  certain  floor,  goes  up  65  ft., 
down  91  ft.,  up  52  ft.,  down  13  ft.,  and  up  65  ft.,  and  stops. 
How  far  and  in  what  direction  is  the  stopping-point  from  the 
starting-point?  Give  your  answer  in  the  form  of  an  alge- 
braic sum. 


Positive  and  Negative  Numbers  75 

9.  A  vessel  starting  in  latitude  +20°  sails  +13°  in  lati- 
tude, then  —60°,  then  +40°,  then  —10°.  What  is  its  latitude 
after  the  sailings  ?  What  is  the  latitude  of  a  ship  starting  in 
latitude  —50°  after  these  changes  of  latitude:  +10°,  —5°, 
+  18°,  -7°,  +38°,  -12°,  +60°?     • 

10.  A  boatman  rows,  at  a  rate  that  would  carry  him  3 
miles  an  hour  through  still  water,  down  a  river  whose  current 
is  2  mi.  an  hour.  What  is  his  rate  per  hour?  What  is  his 
rate  per  hour,  if  he  rows  up  the  river  ? 

Exercise  IX 
Add  the  following,  doing  all  you  can  orally: 

1.  +1  7-   -6|  13-   -i3i  19-   +32&'f: 
±1  ±M                 ±133                   -28b^c 

2.  —I  8.   +7f  14.   —6.69  20.   +iSv''y3       • 
+f  -7i                  +8.04                  -24v'y3 

3-  +U  9-   -i2i        IS-   +  8.95         21.   i-Six-a) 
—  f  —  2^  —11.25  —6{x—a) 

4-  +3i         lo-   ~i8^         16.   —i6r  22.   —i2{x+y) 
-2I               +26i                +i8r  -  7(x+y) 

5.  — 5J         II.   —2.12       17.   —3.25  23.   —6S(m—r) 
-2|                -1.88              -6.85-  -75(m-r) 

6.  +4I         12.   +3.16       18.   +7|^  24.   —6^c—d) 
-6i                -4.08              -65^  +3Kc-c^) 


S'lbtracting  Positive  and  Negative  Numbers 

I.  In  this  problem  positive  numbers  indicate  the  readings 
above  zero  and  negative  numbers,  readings  below  zero.  The 
difference  means  the  number  of  degrees  the  top  of  the  mercury 
column,  must  rise,  or  fall,  to  change  from  the  second  reading 


*5  First-Year  Mathematics 

to  the  first.     If  the  change  is  a  rise,  mark  it  positive  (  +  ),  if 
a  jaU,  mark  it  negative  (— ). 

(I)  (2)  (3)  (4)  (5) 

First  reading:  +68°       -98°       -30°       -  f       +   1° 

Second  reading:        +42°       -18^       +65°       +32°       -28° 

(6)  (7)  (8)  (9)  (10) 

First  reading:  0°  -8°       +6^°       -40°     -  2y° 

Second  reading:        +78°         +8°       -3^°       -ga°     +60/ 

Define  minuend.     Define  subtrahend. 

2.  By  sections  52  and  53  find  the  sums  in  the  following 
problems  and  compare  the  exercises  and  your  results  with 
those  of  the  like  numbered  exercises  of  problem  i : 

(0      (2)      (3)      (4)      (5)       (6)       (7)       (8)       (9)      (10) 
+68    —98    —30    —7+1  o    —8    +6x    —4a    —  2y 

—42    +18    —65    —32    +28    —78    —8    +3^    -\-ga    —6oy 

3.  Show  by  comparing  the  problems  of  i  and  2  that  the 
difference  of  any  two  numbers  can  be  found  by  changing  the 
sign  of  the  subtrahend  and  then  adding. 

4.  Find  the  differences  of  the  following: 

(I)         (2)  (3)  (4)  (5)  (6)  (7) 

+  19       —60       —75       +  8       —3a       +i8x       —i2y 
—10      —25      +25       —16      —2a      —  6x      —  7y 

(8)  (9)  (10)  (II) 

+a        +3^— 18         +  a+26—  c         — i2a+3x 
—b        +2X+  6         +3«— 5^+3^         +  'ja—2x 

54.  To  find  the  difference  of  two  numbers  change  the  sign 
of  the  subtrahend  mentally,  then  add. 


Positive  and  Negative  Numbers  77 

Exercise  X 

Subtract  the  lower  from  the  upper  number  of  the  following, 
doing  all  you  can  mentally: 


I. 

9- 

+  12J 
-i6i 

17- 

-9(c-5) 
+3('^-^) 

2. 

-i 

-i 

10. 

+  8| 

18. 

+ac'^d3s 
—acH^s 

3- 

II. 

—6av' 
+  av' 

19. 

+6.75^3 
-7.2503 

4- 

12. 

-136^3 

+    8^3 

20. 

-3.i6c^cJ 
-o.89c^(i 

5- 

-1^ 
+  i 

13- 

-  gcxy 
-I  sexy 

21. 

-4.76(/+5) 
+9.67(/+^) 

6. 

+4* 
-5i 

14. 

■\-']a¥c 
—gab^c 

22. 

+o.82(a2-x^) 
-3.75(a--x^) 

7- 

-8§ 

+  2i 

IS- 

-Sm^r^s 
+^m^r^s 

23. 

-o.75(c+<i^) 
—  o.9o(c+(/^) 

8. 

-9f 
-3i 

16. 

+6(x+z) 
-7(x+z) 

24. 

+o.9i6a6*c3 
— 1.803  a6'c3 

Multiplying  Positive  and  Negative  Numbers 

Suppose  the  short  spaces  on  the  line  east-west,  E  W,  repre- 
sent a  mile. 

1.  Show  what  space  starting  from  o  in  each  case  repre- 
sents -f-2  mi.;    -F3  mi.;    +5  mi.;    -f  8  mi.;    -f-io  mi. 

\ ,    .    . 

Mi  0  E 

Fig.  71 

2.  Starting  again  from  o,  show  the  space  that  represents 
—  I  mi.;   —2  mi.;   — 5  mi. ;   —7  mi.;    —10  mi.  \ 


jg  First-Year  Mathematics 

3.  Show  the  spaces  from  o  a  man  goes  if  he  travels  +2  mi. 
a  day  for  I'day ;  2  days;  3  days;  5  days. 

4.  What  is  2  times  —  2  mi.  ?  3  times  —  2  mi.  ?  4  times 
—2  mi.?    5  times  —2  mi.? 

5.  What  is  5  times —4  mi.  ?  — lomi.  ?  —  20  mi.  ?  —25 
mi.?     —100  mi:? 

6.  What  is  16  times  —2  mi.?  —5  mi.?  —10  mi.?  —20 
mi.?    —100  mi.  ? 

7.  What  is  5oX(-2)?  25X(-io)?  4oX(-i2)? 
8X(-i2o)?    4X(-a)?     ioX(-^)? 

8.  The  value  of  a  man's  property  changes  by  +$1000  a 
year.  How  much  does  it  change  in  2  yr.  ?  5  yr.  ?  8  yr.  ? 
10  yr.  ?  In  each  case  tell  whether  the  change  is  an  increase, 
or  a  decrease. 

9.  If  the  value  of  a  man's  property  changes  by  —$500  a 
year,  how  much  does  it  change  in  3  yr.  ?  5  yr.  ?  7  yr.  ?  8  yr.  ? 
10  yr.  ?  12  yr.  ?  In  each  case  tell  whether  the  change  is  an 
increase,  or  a  decrease. 

10.  How  much  and  in  what  way  does  a  man's  property 
change  in  12  yr.  at  the  rate  of  +$50  a  yr.  ?  — $100  a  yr.  ? 
+$400  a  yr.  ?     +$1,800  a  yr.  ?     —$2,000  a  year  ? 

11.  How  much  and  in  what  direction  does  the  height  of  a 
mercury  column  of  a  thermometer  change  in  6  hours  at  the 
rate  of  +10°  an  hr.  ?  -8°  an  hr.  ?  -7°  an  hr.  ?  +5^°  an 
hr.  ?    -3i°  an  hr.  ?    +2^°  an  hr.  ?    -a°  an  hour  ? 

12.  State  a  way  of  multiplying  a  negative  number  by  an 
arithmetical  number. 

55.  A  light  bar  (Fig.  72)  supplied  with  equally  spaced  pegs 
is  balanced  about  its  middle  point,  M.  With  a  number  of 
equal  weights,  w,  the  following  experiments  are  performed. 


Positive  and  Negative  Numbers 


79 


1.  Hang  a  weight  of  2W  on  the  peg  l,.  This  weight  tends 
to  turn  the  left  end  of  the  bar  downward.     How  much  weight 

must   be   attached  to  the     ,,„,,„„„„„„„„„ ,„„„ 

hook,  H,  to  balance  this 
turning-tendency  ?  Now 
hang  the  same  weight,  2iv, 
on  peg  I2,  and  measure 
its  downward  turning-ten- 
dency by  attaching  to  the 
hook,  H,  a  weight  suflScient 
to  balance  the  bar. 

2.  In  a  similar  manner 
find  the  turning-tendency 
caused  by  the  weight  2W  on 
the  peg  I3;  on  1^;  on  I5. 

3.  Using  a  weight   of 
3W,   find   the  turning-ten- 
dency when  placed  on  peg  ^^^-  7^ 
Ij-,  on  Ij;  on  I3;  on  1^;  on  I5. 

4.  Perform  experiment  3,  using  a  weight  4W. 

56.  Experiments  i,  2,  3,  and  4  show  that  when  a  weight 
(as  2w)  is  hung  on  peg  Ij,  the  turning-tendency  caused  by  this 
weight  (21/;)  is  five  times  as  great  as  when  the  same  weight  is 
hung  on  peg  l^;  on  1^  its  turning-tendency  is  four  times  as 
great  as  on  1,;  on  I3  it  is  three  times,  and  on  I,  it  is  two  times 
as  great  as  on  1,. 

57.  The  same  facts  hold  when  ^w,  4W,  or  any  other  weight 
is  used.  In  other  words,  with  an  apparatus  like  that  in 
Fig.  72,  the  turning-tendency,  or  leverage,  caused  by  a  weight 
is  measured  by  the  product  0}  the  weight  by  the  distance 
from  the  turning-point,  M,  to  the  peg  where  the  weight 
hangs. 


8o  First-Year  Mathematics 

1.  What  is  the  turning-tendency  caused  by  a  weight  of 
jw  hung  on  peg  Ij  ?    on  I3  ?    a  weight  of  qw  on  I4  ?    on  1,  ? 

2.  If  the  bar  had  12  pegs  on  each  side  of  M,  what  would 
be  the  turning-tendency  caused  by  a  weight  of  8w  on  peg  1„? 
yw  on  I9?    w  on  l,,  ? 

3.  Attach  the  cord,  C  (Fig.  72),  to  peg  r^  and  perform  the 
experiments  of  §55  on  the  right  side  of  the  bar. 

58.  Experiment  3,  §57,  shows  that  the  same  facts  are  true 
on  the  right  side  as  on  the  left,  but  the  bar  turns,  or  tends 
to  turn,  in  the  opposite  direction.  To  avoid  confusion,  some 
simple  method  of  distinguishing  between  these  two  directions 
of  turning  is  desired. 

Suppose  a  watch  laid  upon  the  page  of  the  book  with  its 
face  up.  When  the  bar  turns,  or  tends  to  turn  Tvith  the  hands 
of  the  watch,  the  turning-tendency  will  be  called  negative 
and  designated  — ;  if  it  turns,  or  tends  to  turn,  against  (oppo- 
site to)  the  watch-hands,  the  turning-tendency  will  be  called 
positive  and  designated  +. 

59.  In  all  the  experiments  thus  far  performed  the  weights 
which  caused  the  bar  to  turn  were  downward-pulling  weights, 
or  forces.  By  arranging  an  apparatus  as  in  Fig.  73,  p.  82, 
forces  can  also  be  made  to  pull  upward.  Downward-pulling 
weights  or  forces  will  be  designated  by  — ,  and  upward-pulling 
forces  by  -h. 

60.  The  distance  from  the  turning-point,  M,  to  the  peg 
where  the  weight,  or  force,  acts  will  be  called  the  lever-arm, 
or  arm,  of  the  force.  Lever-arms  measured  from  the  turning- 
point  toward  the  right  will  be  marked  -1-;  those  toward  the 
left,  -. 

For  example,  if  the  distance  from  M  to  peg  r,  be  repre- 
sented by  -I- 1,  then  the  distance  from  M  to  the  peg  r^  will  be 
represented  by  -f  4;  from  M  to  I3,  by  —3,  and  so  on. 


Positive  and  Negative  Numbers 


81 


61.  If  a  force  of  —2  acts  on  peg  I3,  its  turning-tendency, 
or  leverage,  is  the  product  of  —2  and  —3.  Since  the  bar 
tends  to  turn  against  the  watch- hands,  the  turning-tendency 
is  written  +6.     If  T"  represents  the  turning-tendency,  then 

T={-2)- (-3)^-^6. 

If  a  force  of  —2  acts  on  r^,  its  turning-tendency  is  the 
product  of  —2  and  +3,  or  —  6,  since  the  bar  tends  to  turn 
with  the  watch-hands.     In  this  case 

T={-2)-(+3)  =  -6. 

I.  In  the  following  experiments  write  (as  in  §61)  the 
leverages,  or  turning-tendencies,  for  the  forces  and  arms  in- 
dicated : 


Forces 

Arms 

Leverages 

Forces 

Arms 

Leverages 

I 

-3 

—  2 

VII 

-6 

+    I 

II 

-4 

+  2 

VIII 

-8 

—    I 

III 

-7 

-3 

IX 

—  a 

+     b 

rv 

-6 

-4 

X 

—  m 

—     z 

V 

—  I 

+  5 

XI 

-3W 

—    2X 

VI 

-8 

+9 

XII 

-6c 

-i2y 

2.  If  loadings  I  and  VII  in  the  foregoing  tables  were  on 
the  apparatus  at  the  same  time,  would  the  bar  balance  or 
turn  ?  If  it  turns,  in  what  direction  would  it  turn  ?  Answer 
similar  questions  for  II  and  VIII;  V  and  VII;  VI  and  VIII. 

62.  By  the  apparatus  in  Fig.  73  (p.  82)  forces  can  be 
made  to  pull  upward  on  either  side  of  the  bar. 


82 


First-Year  Mathematics 


tl'llKlllllltltll 


I,  t.  I,   U  I. 


I.  M   r.    n  n 


If  a  force  of  2  pulls  upward  on  peg  r^,  its  turning-tendency 
is  the  product  of  +2  and  +4.  Since  the  bar  tends  to  turn 
against  the  watch-hands,  the 
turning-tendency  is  written 
+8.  Using  T  for  the  turning- 
tendency, 

r=(+2)(-H4)=+8. 

On  peg  I4  the  turning- 
tendency  is  the  prcduct  of 
+2  and  —4,  or  —8,  since 
the  bar  tends  to  turn  ^vith 
the  watch  hands.  In  this 
case 

r=(+2)(-4)=-8. 

1.  In  the  following  experi- 
ments write  the  leverages  for 
the  indicated  arms  and  forces : 


Fig.  73 


Forces 

Arms     Leverages 

Forces 

Arms 

Leverages 

I 

+3 

—  2 

V 

+6 

+    I 

II 

+7 

-3 

VI 

+  m 

+  7« 

III 

+  1 

+3 

VII 

+3 

—  2 

IV 

+8 

+9 

VIII 

-|-6w 

—  I2Z 

2.  If  III  and  VII  in  the  foregoing  tables  are  on  the  appara- 
tus at  the  same  time,  will  the  bar  balance  or  turn  ?  If  it  turns, 
in  which  direction  will  it  turn  ?  Answer  similar  questions  foi 
landV;  II  and  VII;  IV  and  VII. 


3.  Write  the  leverages  for  the  following  loadings: 


Positive  and  Negative  Numbers 


83 


Forces 

Arms 

Forces 

Arms 

I 

+8 

-9 

VII 

+  7 

+8 

II 

+8 

+9 

VIII 

-7 

+8 

III 

-8 

+9 

IX 

+  7 

-8 

IV 

-8 

-9 

X 

-7 

-8 

V 

+    X 

-ly 

XI 

—  a 

+6& 

VI 

+4^ 

+zy 

XII 

-5« 

-2& 

4.  What  sign  (direction)  has  the  turning-tendency  of  a 
plus  force  with  a  plus  arm  ?  A  plus  force  with  a  minus  .arm  ? 
A  minus  force  with  a  plus  arm  ?  A  minus  force  with  a  minus 
arm? 

63.  If  3(~4)  nieans  that  four  is  to  be  measured,  or  laid 


-12     -8 


-♦        0 

Fig.  74 


off,  3  times  from  o  in  the  negative  direction  (the  direction  of 
—4),  what  is  the  value  of  the  product,  3  (—4),  or,  what  is  the 
same  thing,  (+3)(-4)? 

The  product  means  the  distance  and  direction  from  o,  evidently  — 12. 

1.  Show  on  a  figure  the  meaning  and  value  o^  the  follow- 
ing products: 

(i)  2(+3)  (3)  3(-2)  (5)  2,{-S)  (7)  2(+6) 

(2)  2(-3)  (4)  3(+2)  (6)  i(+5)  (8)  2(-6). 

2.  Show  on  a  figure  the  products:   (— 3)(+4);   (— 3)(— 4)- 
(  —  3)  (4- 4)  means  that  four  is  to  be  laid  ofiF  3  times  in  the  direction 

opposite  to  the  direction  of  4-4,  i.  e.,  in  the  negative  direction. 

(  —  3)  (  —  4)  means  that  four  is  to  be  laid  off  3  times  in  the  direction 
opposite  to  the  direction  of  —4,  i.  e.,  in  the  positive  direction. 


84  First-Year  Mathematics 

3.  Interpret  these  products  on  the  same  principle: 

(1)  (-2)(-3)  (4)  (+2)(+8)  (7)  (-2)(  +  S) 

(2)  (+3)(-2)  (5)  (-3)(-5)  (8)  (-2)(-S) 

(3)  (-2)(+4)  (6)  (+3)(-5)  (9)  (-3)(+6). 

4.  Show  on  Fig.   75  the  value  of  2  times   +a;    2(— a), 
(-2)(+a);    (-2)(-a);    (-3)(-^);    (-3)(+«);    (-4)(-«)- 


-4a  -3o  -2a    -a     0     a  2a    3a  ia 

Fig.  75 

5.  Interpret  the  meanings  of: 

(i)  (+3)(+«)  (S)  (+«)(+&)  (9)  (+c)(-d) 

(2)  (+3)(-«)  (6)  (+a)(-&)  (10)  i+c){-\-d) 

(3)  (-3)(+^)  (7)  (-«)(+&)  (")  (-0(+^) 

(4)  (-3)(-^)  (8)  (-«)(-&)  (12)  (-0(-^)- 

SUMMARY 

In  digits:  In  letters: 

■     (+3)(+4)  =  +i2  (+a)(+6)  =  +a6 

(+3)(-4)  =  -i2  (-fa)(-6)  =  -a6 

(-3)(+4)  =  -i2  (_a)(+6)  =  -a6 

(-3)(-4)  =  +i2  (-a)(-6)  =  +a&. 

6.  Examine  the  eight  products  of  the  summary  and  make 
a  rule  for  obtaining  the  algebraic  sign  of  a  product  of  two 
numbers  from  the  signs  of  the  factors. 

Compare  your  rule  with 

64.  Law  of  Signs  for  Multiplication: 

//  two  factors  have  like  signs,  tlie  product  is  positive;  if 
two  factors  have  unlike  signs,  the  product  is  negative. 

Exercise  XI 

Find  the  products  of  the  following,  doing  all  you  can  men- 
tally: 


Positive  and  Negative  Numbers 


85 


I 

(+§)(  +  i) 

16.  ( 

:-3m+xy) 

2 

(-§)(  +  f) 

17. 

[-5*)(-^z) 

3 

(  +  l)(-|) 

18.  ( 

-si«)(+&) 

4 

(-f)(+l) 

19.  ( 

-2ic)(-2irf) 

5 

(-J)(-A) 

20.  1 

+  7i^)(-7i>') 

6 

(  +  T\)(-f) 

21.  ( 

:+5)(-^^^) 

7 

(-6f)(-6|) 

22. 

:-7)(-ro 

8 

(+6i)(-6f) 

23-  < 

'-lo)(+7;='j(;) 

9 

(-6f)(  +  6f) 

24. 

;  +  i2)(— 03/3) 

10 

(+6|)(  +  6f) 

25- 

:-i6)(+ak^) 

II 

i  +  3)i-ad) 

26. 

[  +  bc)(-a) 

12 

(-3)(-'^^) 

27.  ( 

+v''x)(-a''b) 

13 

(-7)(+^^) 

28.  ( 

-\-X')(-X3) 

14 

(-«)(+/'?) 

29.  ( 

'+y^)i-x-) 

15 

("OC-/"-) 

30-  ( 

+  6r^)(-ir»). 

Dividing  Positive  and  Negative  Numbers 

65.  The  product  of  two  factors  and  either  factor  being 
known,  the  other  factor  is  the  quotient  arising,  as  in  arithme- 
tic, from  dividing  the  known  product  by  the  known  factor. 

66.  The  quotient  of  a  6  divided  by  b  is  indicated  thus, 


ab-i-b,  or 


ab 


oi  ab  divided  by  —a,  thus,  ab-^(—a),  or 


ab 


I.  Since   (  +  2)(  — 5)  =  — 10,   what    must    — Io-^(+2), 


or 


•10 


be? 


—  10 


+  2  -5 

2.  Looking  at  the  first  four  equations  in  the  foregoing 
summary,  §§63  and  64,  answer  the  following  questions,  giving 
reasons  for  answers: 

(1)  (  +  i2)-(+3)=?  (5)  (-I2)-^(-3)=? 

(2)  (  +  i2)-(  +  4)=?  (6)    (-i2)H-(+4)=? 

(3)  (-i2)4-(+3)=?  (7)  (  +  i2)^(-3)=? 

(4)  (-i2)-(-4)=?  (8)  (  +  i2)-(-4)=? 


86  First-Year  Mathematics 

3.  From  the  second  list  of  four  equations  in  the  foregoing 
summary  answer,  with  reasons,  the  following: 

(i)  (+ab)^{+a)=?  (5)  (-aft)-^(-a)=? 

(2)  (+a6)^(+6)=?  (6)  (-a6)^(+&)=? 

(3)  {-ab)H+a)=?  (7)  (+a6)-J-(-a)=? 

(4)  {-ab)^{-b)=?  (8)  (+ab)^(-b)=? 

4.  Answer  the  following,  with  reasons: 
,)±|=,  (3)±^=p  (,)2=, 

(.)±^«=,  (6)±|=?  1-)^^=? 

,  .    -18     -           .  .    -18     ^          ,     .    -xy 
(7)  =?  (7)  =?  (11)  -=? 

(4)^='     («)^=?     (")^^-? 

5.  Examine  the  answers  to  problems  2  and  3  and  state 
what  the  sign  of  the  quotient  is 

(i)  if  the  sign  of  both  dividend  and  divisor  is  plus,  (+); 

(2)  if  the  sign  of  both  div^idend  and  divisor  is  minus,  (— ); 

(3)  if  the  sign  of  the  dividend  is  plus,  (  +  ),  and  of  the  di- 
visor, minus,  (— ); 

(4)  if  the  sign  of  the  dividend  is  minus,  (— ),  and  of  the 
divisor,  plus,  (+); 

(5)  if  the  signs  of  dividend  and  divisor  are  alike  (i.  e., 
both  +,  or  both  — ); 

(6)  if  the  signs  of  dividend  and  divisor  are  unlike  (i.  e., 
one  —  and  the  other  +). 

6.  State  the  law  of  signs  for  division  and  compare  the 
statement  with 

67.  Law  of  Signs  for  Division: 

//  dividend  and  divisor  have  like  signs  the  qitotient  is  posi- 
tive; if  dividend  and  divisor  have  unlike  signs  the  quotient  is 
negative. 


Positive  and  Negative  Numbers 


87 


Answer  the  following: 


] 

+  1 
'■    -t 

'-=? 

4- 

a 

6a      . 

2 

-5 

5- 

—a 

8.  -«"-? 

a 

•  +10     * 

6. 

m 

—  i2ax     - 

n                           —  1* 

^ 

9.                  —  r 

12a 

Exercise  XII 

Find  the  quotients  of  the  following,  doing  all  you  can  1 

tally: 

I. 

[+!)- 

-(+*) 

21. 

{+a)^(-{a) 

2. 

[-!)- 

-(+!) 

22. 

[-a3)-(+a) 

3- 

[H)- 

-(-!) 

23- 

(+a^)^(-a^) 

4- 

[n)- 

-(+f) 

24. 

:-a4)--(+a3) 

5- 

:+f)- 

-(-f) 

25-  < 

;-6a3)^C+3a) 

6. 

:+!)- 

-(-§) 

26.  ( 

:+4ab)^(-b) 

7- 

^-f)- 

-(+f) 

27.  ( 

'+4ab)-r-(—2a) 

8.  ( 

:+!)- 

-(-i) 

28.  ( 

-Sax)^(+Sx) 

9- 

:-f)- 

-(-1) 

29. 

[  —  2xy)-7-{-\-xy) 

10. 

:-i)- 

-(+1) 

30-  < 

:  +  6fr^)-(-3Jr) 

II.  ( 

:-f)- 

-(-f) 

31-  < 

[-{-av^)-i-{—av) 

12.  ( 

'+f)- 

-(+f) 

32.  ( 

-av3)^{-\-v^) 

13-  ( 

-f)- 

-(-f) 

33-  ( 

—abc)-i-(—a) 

14.  ( 

-f)- 

-(+f) 

34-  ( 

'+abc)-i-(+ac) 

IS-  ( 

.+  H)-(-|) 

35-  ( 

—abc')^{—ac) 

16.  ( 

-2a)--(+2) 

36.  ( 

'—avr)-^{+ar) 

17-  ( 

-2a)-=-(-a) 

37-  ( 

+  7fax)4-(+23rt) 

18.  ( 

-5^)-(+5) 

38.  ( 

-7M^(-^^b) 

19.  ( 

-\-l2X)^{-4X) 

39-  ( 

-I6b■z3)-^(+4i>'z) 

20.  ( 

-a)-i 

-(-ia) 

40.  ( 

+6.82flz)^(-3ia). 

88  First-Year  Mathematics 

Summary 

Positive  and  negative  numbers  are  needed  to  record  tem- 
perature, directed  distances,  opposing  forces,  opposite  latitudes, 
debts  and  assets,  dates  before  and  after  the  beginning  of  the 
Christian  era,  motions  in  opposite  directions,  and  directed 
magnitudes. 

Graphs  may  be  made  to  represent  thermometer  readings, 
average  temperatures,  ages  and  heights  of  persons,  batting 
averages,  populations,  rain  and  snowfall,  lengths  of  day,  areas 
of  figures,  equations,  and  general  expressions  of  number. 

The  algebraic  sum  of  nvmibers  with  like  signs  is  the  arith- 
metical sum,  with  the  common  sign  prefixed. 

The  algebraic  sum  of  two  numbers  with  unlike  signs  is 
the  arithmetical  difference,  with  the  sign  of  the  larger  nvunber 
prefixed. 

The  algebraic  sum  of  any  number  of  numbers  may  be 
found  (i)  by  adding  all  the  numbers  in  order,  or  (2)  by  adding 
all  the  positive  numbers,  then  all  the  negative  numbers,  then 
adding  the  sums. 

The  algebraic  difference  of  two  numbers  is  found  by  add- 
ing to  the  minuend  the  subtrahend  iiith  sign  changed. 

The  turning-tendency,  or  leverage,  of  a  force  on  a  bar  or 
lever,  is  the  product  of  the  force  by  the  lever-arm. 

If  two  factors  have  like  signs  the  product  is  positive.  If 
two  factors  have  unlike  signs  the  product  is  ttegative. 

If  dividend  and  divisor  have  like  signs  the  quotient  is 
positive. 

If  dividend  and  divisor  have  unlike  signs  the  quotient  is 

negative. 


CHAPTER  V 


BEAM  PROBLEMS  IN  ONE  AND  TWO  UNKNOWNS 

Problems  in  One  Unknown  Number 

68.  In  this  chapter  some  practical  problems  arising  out  of 
the  common  uses  of  forces  will  be  solved  by  means  of  the 
equation.     It  is  necessary  first  to  discover  a  law  of  these  forces.* 


* 6- 

I   1   I   I   I 


•»3 


I  I  I   I  I  I  I 


<~t%^-* 


T 


XI 


-i2 


Fig.  76 


Fig.  77 


I.  A  bar  or  lever  (Fig.  76)  has  loadings  as  in  Fig.  78. 
Draw  a  diagram  (Fig.  77)  and  find  the  turning-tendency, 
or  leverage,  for  each  loading: 


No. 

Force 

Arm 

No. 

Force 

Arm 

I 

+3 

-6 

VII 

-    4 

-9 

II 

+  2 

-3 

VIII 

-   3 

—  2 

III 

+3 

+  2 

IX 

—  12 

+3 

IV 

X 

+3 

X 

X 

-3 

V 

X 

—  2 

XI 

X 

+  2 

VI 

+3 

X 

XII 

-     2 

X 

Fig.  78 

2.  Draw  a  diagram  showing  I  and  VIII  (Fig.  78)  on  the 
bar  at  the  same  time;   I  and  VII;   II  and  VIII;   II  and  IX; 

*  See  pages  78-83,  on  turning-tendencies. 
89 


90 


First- Year  MatJiematics 


I,  III,  and  VIII.     In  each  case  state  whether  the  bar  balances 
or  turns,  and  if  it  turns,  in  which  direction  it  turns. 

69.  In  problem  2,  if  the  turning-tendencies  for  I  and  VIII 
are  added,  the  total  turning-tendency  is 

(+3)  •  (-6)  +  (-3)  •  (-2)  =  (-i8)-h(+6)  =  -i2, 
which  says  in  mathematical  language  that  the  bar  does  not 
balance,  but  turns  in  the  negative  direction. 

If  the  turning-tendencies  for  II  and  VIII  are  added,  the 
total  turning-tendency  is 

(+2)  •  (-3)  +  (-3)  •  (-2)  =  (-6)-h(+6)=o, 

which  shows  that  the  bar  does  not  turn. 

If  the  turning-tendencies  for  III,  VII,  and  IX  are  added, 
the  total  turning-tendency  is 

(+3)(+2)(-4)(-9)  +  (-i2)(  +  3)  = +6+36-36= +6, 

which  shows  that  the  bar  does  not  balance,  but  turns  in  the 
positive  direction. 

70.  If  two  or  more  forces  are  acting  on  the  bar  at  the 
same  time,  the  total  turning-tendency  is  found  by  adding  alge- 
braically the  separate  turning-tendencies.  If  the  algebraic  sum 
is  zero,  the  bar  balances.  If  the  sum  is  not  zero,  the  bar 
turns  in  the  direction  indicated  by  the  sign  of  the  sum. 

I.  Find  the  total  turning-tendency,  and  interpret  it,  as  in 
§69,  when  the  following  loadings  of  Fig.  78  are  on  the  bar  at 
the  same  time.     Draw  a  diagram  for  each  case: 

(i)  I  and  IX  (3)  I,  III,  and  IX 

(2)  II  and  VII  (4)  I,  III,  and  VII. 


-.»-_ 


ZST 


.__  2.  If  a  force  of  —4  on  an 

arm   of    —9    (Fig.    79),   and    a 
♦/  force  of  -f-/  on  an  arm  of  —2,  are 

Fio.  79  on  the  bar,  what  must  /  be,  for 

balance  ? 


Beam  Problems  in  One  and  Two  Unknowns 


91 


If  the  bar  is  balanced,  the  sum  of  the  turning-tendencies  must  be 
zero.     We  may  then  write 

(/)(-2)  +  (-4)(-9)=o.  (i) 

Multiplying  —2/+ 36  =  0.  (2) 

Subtracting  36  —  2/=  —  36.     What  axiom  ?  (3) 

Dividing  by  —2  /=     18.     What  axiom?  (4) 

Check:  From  equation  (i),  (  +  i8)(  — 2)-|-(— 4)(  — 9)  =  (  — 36)-|- 
(+36)=o. 

3.  Write  the  equation  of  total  turning-tendency,  and  find 
what  the  unknown  force  or  arm  must  be,  for  balance,  for  each 
of  the  five  loadings  of  Fig.  80. 


No. 

Force 

Arm 

Force 

Arm 

Force 

Arm 

I 

/ 

+  3 

+3 

-6 

II 

+3 

-6 

w 

—  2 

III 

r 

—2 

r 

-3 

-4 

-9 

IV 

—  2 

^  d 

d 

+3 

+  2 

-3 

V 

—  2 

I 

+  3 

/ 

+  3 

+  2 

.^. 


Fig.  So 

71.  Law  of  Turning-Tendencies  or  Leverages.     For  balance, 
the  algebraic  sum  of  all  the  turning-tendencies  must  equal  zero. 

I.  A  bar  (Fig.  81)  is  balanced 
by  a  force  of  -l-io  on  an  arm  of    q- 
—6,  and  a  force  of  5  +  3  on  an 
arm  of  -H5.     Find  the  values  of 
5  and  5 -[-3.  YiG.  81 

»--4--*<^+J-* 


~T — Ji-T 

ami-is  *n 

Fig.  82 


2.  A  force  of  3 w— 15  (Fig.  82) 
on  an  arm  of  —4  is  balanced 
by  a  force  of  -f  12  on  an  arm  of 


+3.    Find  the  values  of  w  and  3W— 15. 


93 


First-Year  Mathematics 


3.  A  bar  is  balanced  by  each  of  the  five  loadings  of  Fig.  83. 
Draw  a  sketch  and  find  the  value  of  the  unknown  jEorces  and 
arms  for  each  loading. 


No. 

Force 

Arm 

Force 

Arm 

Force. 

Arm 

I 

W+   5 

-3 

+15 

+4 

II 

t-  3 

+7 

-/ 

-8 

+  13 

-3 

III 

+  3 

w-s 

+  15 

-4 

IV 

+  3 

-8 

+  zk 

+4 

—  2 

+4* 

V 

+39I 

-4 

5^ 

+  2i 

-i3i 

-4 

Fig.  83 


Practical  Applications 

I.  A  B  (Fig.  84)  is  a  crowbar,  6\  ft.  long,  supported  at  F, 
\  ft.  from  A.  A  stone  presses  down  at  A  with  a  force  of 
1,800  pounds.  How  many  pounds  of  force  must  be  exerted  by 
a  man  pressing  down  at  B  to  raise  the  stone  ? 


£^ 


P 1 


Fig.  84 

2.  With  other  conditions  as  in  problem  i,  what  would  be 
the  pressure  at  B  if  the  fulcrum  F  (point  of  support)  were  3 
in.  from  A  ? 

3.  With  the  fulcrum  \  ft.  from  A  (Fig.  84)  what  weight 
would  be  held  in  balance  by  a  pressure  of  200  lb.  at  B  ? 


Beam  Problems  in  One  and  Two  Unknowns 


93 


4.  A  suction  pump  (Fig.  85)  is  a  device  for  raising  water 
from  wells.  The  handle,  O  B, 
works  against  a  pin  at  A,  so 
that  when  the  hand  pushes 
downward  at  B,  the  point  O 
rises,  and  by  the  aid  of  a  piston 
on  the  lower  end,  C,  raises 
a  mass  of  water.  If  OA  =  2 
in.  and  OB  =3  ft.,  what  load 
at  O  will  be  raised  by  a  force 
of  20  lb.  pushing  downward 
at  B? 

5.  With  other  conditions  as 
in  4,  what  force  will  be  exerted 
at  O  by  a  downward  force  of  68 
lb.  at  B  ? 

6.  A  stone   slab   S   (Fig.   86),  weighing   2,400   lb.,   rests 
with  its  edge  on  a  point  B,  6  in.  from  the  fulcrum  F  of  a  crow- 
bar F  A,  6  ft.   long.     How  many  pounds  of 
force  must  be  exerted  at  A  to  raise  the  slab  ? 


?4 

i 

■ 

— *— - 

» 

Fig.  8s 


6' ^ 


L 


Fig.  86 


— 14- -> 

-966  ^ 

Fig.  87 


7,  A  steel  beam  24  ft.  long  and  weighing  966  lb.  (Fig.  87) 
is  being  moved  by  placing  under  it  an  axle  borne  by  a  pair  of 
wheels,  as  shown  at  A,  the  end  B  being  carried.  If  the  axle  is 
2  ft.  from  the  middle  of  the  beam,  what  is  the  weight  at  B  ? 

The  weight  of  the  beam  itself  may  be  treated  as  a  load  of  966  lb. 
hanging  to  the  bar  at  the  middle  point. 


94 


First- Year  MatJiematics 


8.  K  the  supporting  point  in  problem  7  had  been  11  ft. 
from  O,  what  force  at  B  would  have  balanced  the  rail  ? 

9.  How  far  must  the  axle  in  problem  7  be  placed  from 
the  middle  M,  that  the  weight  at  B  may  be  241 J  poimds 
(Fig.  88)  ? 

-u'- 


^ 


-96  J 

Fig.  88 


-fMlVl 


10.  With  the  fulcrum  2  ft.  to  the  left  of  M,  what  would  be 
the  weight  of  the  rail  if  it  is  balanced  by  an  upward  force  of 
140  lb.  at  B  ? 

11.  How  may  a  steel  rail  weighing  more  than  a  ton  be 
weighed  with  a  pair  of  balances  reading  only  to  60  pounds? 

12.  A  dry  goods  box  (Fig.  89)  weighing  360  lb.  is  being 
moved  along  the  floor  by  the  aid  of  a  roller.  If  the  box  is 
6  ft.  long,  what  force  at  C  is  needed  to  hold  the  box  horizontal 
when  the  roller  is  i  ft.  from  B  ?    2  ft.  from  B  ? 


rt.-r--6' 


1      t 

Fig.  89 


Fig.  90 


13.  A  wheelbarrow  (Fig.  90)  is  loaded  with  45  bricks, 
averaging  6  lb.  a  piece.  What  lifting  force  will  be  needed  at 
A  to  raise  the  load  if  the  bar  O  A  is  4^  ft.  and  the  distance 


Beam  Problems  in  One  and  Two  Unknowns  95 

t 

from  the  center  of  the  wheel,  O,  to  the  point,  B,  where  the 

vertical  line  through  the  center  of  the   load   crosses  A  O,  is 
2  feet  ? 

14.  With  the  same  load  and  length  of  bar,  O  A,  as  in 
problem  13,  how  far  is  it  from  O  to  the  crossing-point  B  of 
the  vertical  center  line  of  the  load,  if  90  lb.  at  A  just  raises  it  ? 

Exercise  XIII 
Simplify  the  following  products  and  quotients: 

1.  (-i5-0(+3)  J4    -3^xy 

2.  (io-0(-85)  '     +4:y 
3-  (30(7-0                 IS.  IH^ 

4.  (w-4)(+8i)  ^ 

5.  (-i5-70(-6)        16. 

6.  (-i)(-5^) 
7-  (-i)(-S^  +  7)        17- 
8.  (  +  i)(-5^  +  7) 

9-  (  +  5)(o)  ^8.  — -^-^ 

10.  (-s)(o)  o 

11.  (o)(  +  342)  '9-  :^ 

12.  (o)(-763)  o 

20   — 

13.  {-2k){l+X)  '    -f 

72.  It  is  often  convenient  to  measure  lever-arms  from  a 
point  at  which  the  bar  is  not  actually  supported. 

I.  A  basket  weighing  56  lb.  hangs  on  a  stick  8  ft.  long 
(Fig.  91)  at  a  point  i  ft.  from  the  middle,  while  it  is  being 

•«■ 4' »•< 4' ► 

M. 


-gy 
+  1 
-15^  +  7 


♦S«  -56  ♦Jx 

Fig.  91 
carried  by  two  boys,  one  at  each  end.     The  boys  lift  ^x  and 
TfX  lb.,  respectively.     Find  the  values  of  x,  e^x,  and  t,x. 


96  First- Year  MathenuUics 

The  bar  or  stick  is  balanced,  no  matter  what  point  on  the 
bar  is  regarded  as  the  turning-point,  or  fulcrum.  The  lever- 
arms  may,  therefore,  be  measured  from  any  point  on  the  bar, 
whether  the  bar  is  actually  supported  at  that  point,  or  not. 

Taking  M  as  the  t\iming-point,  the  lever-arms  are 
-4,    -I,    +4, 
and  the  turning-tendencies  are 

(+5^)(-4),    (-56)(-i),   (  +  3*)(+4). 
Since  the  bar  does  not  turn,  the  sum  of  these  turning-tendencies  must 
equal  zero. 

(  +  5^)(-4)  +  (-56)(-i)-l-(  +  3»)(+4)=o. 
Solve  the  equation. 

It  should  be  kept  in  mind  that  once  a  certain  point  is 
selected  from  which  to  measure  lever-arms,  all  lever-arms 
must  be  measvured  from  this  point  throughout  the  solution. 

< 6' >< 6* >  ^ 16' ^ 

Fig.  92  Fig.  93 

2.  A  bar  is  balanced  by  the  forces  shown  in  Figs.  92  and 
93.  Find  the  values  of  the  unknown  forces,  measuring  all 
lever-arms  from  M. 

^ , —^  L     « 

Fig.  94  Fig.  95 

3.  Find  the  values  of  the  unknown  forces  on  a  bar  balanced 
as  shown  in  Figs.  94  and  95.  Measure  all  lever-arms  from 
the  end-point  L. 

4.  Solve  problem  3,  measuring  all  lever-arms  from  the  end- 
point  R. 


Beam  Problems  in  One  and  Two  Unknowns 


97 


Problems  in  Two  Unknowns 

73.  Beam  or  lever  problems  in  which  two  numbers  are 
unknown  also  may  be  solved  by  the  aid  of  the  law  of  lever- 
ages. 

<- J' -♦«.  —  3' 

A M 

f 


t[ 


+J 


Fig.  96 


1.  A  basket  weighing  24  lb.  (Fig.  96)  hangs  on  a  stick 
6  ft.  long,  at  a  point  i  ft.  from  the  middle,  while  it  is  being 
carried  by  two  boys,  A  and  B.     How  much  does  each  boy  lift  ? 

Using  M  as  turning-point,     ^y  —  24  —  ^x  =  o.     Why?  (i) 

y=8+x.     Why?  (2) 

2.  Can  we  find  from  the  foregoing  equation  (2)  the  values 
of  X  and  y  that  satisfy  problem  i  ?  To  study  this  question, 
copy  and  fill  in  the  following: 

In  the  equation  y = 8 + » 


(2) 


iix=  I,  then  y=8-|-i  =9 
U  x  =  .2,  then  y=8  +  2  =  io 
i£x=  5,  theny  =  8  +  5  =  i3 
iix=  8,  theny=8  +  8  =  i6 
U  x—io,  then  y  =  8  + 
if  «  =  ii,  then  y  = 
etc. 


Fig.  97 
3.  Show  that  each  pair  of  values  of  x  and  y  in  Fig.  97 
satisfies  equation  (2)  of  problem  i. 


X 

y 
9 

I 

2 

10 

5 

13 

8 

16 

ID 

II 

14 

19 

q8  First-Year  Mathematics 

4.  Show  that  many  other  pairs  of  values  of  x  and  y  can 
be  found  as  those  in  Fig.  97  were  found. 

From  2,  3,  and  4  it  follows  that  we  cannot  tell  from  equa- 
tion (2),  done,  which  pair  of  values  in  Fig.  97  gives  the  weights 
that  A  and  B  lift. 

If  B  lifts  9  times  as  much  as  A,  then  the  pair  9  and  i 
(Fig.  97)  satisfies  problem  i. 

If  B  lifts  5  times  as  much  as  A,  then  the  pair 
10  and  2  satisfies  problem  i. 

To  find  the  weights  we  must,  therefore,  know 
a  second  relation  between  them. 
5.  From  Fig.  96,  p.  97,  show  that 

x^ry=2^.  (3) 

Make  a  table  (Fig.  98)  of  ten  pairs  of  values  of 
X  and  y,  that  satisfy  equation  (3).      For  conven- 
ience, solve  equation  (3)  for  y,  thus 
PiQ  gg  y  =  2^—x.     What  axiom?  (4) 

6.  Which  pair  of  numbers  in  Fig.  98  is  also  in  Fig.  97  ? 
Show  that  these  numbers  satisfy  equation  (3),  problem  5,  and 
equation  (2),  problem  i,  and  that  they  are  the  numbers  of 
pounds  that  A  and  B  lift.  Does  any  other  pair  of  numbers 
in  either  table  satisfy  both  equations  ? 

7.  Make  a  table  of  pairs  of  values  of  R  and  S  that  satisfy 
the  equation 

R-2S=%.  (i) 

For  convenience,  solve  equation  (i)  for  R,  thus 

R=2>  +  2S.  (2) 

8.  Make  a  table  of  pairs  of  values  of  R  and  5  that  satisfy 
the  equation 

R+S=i^.  (3). 


X 

y 

4 

20 

5 

19 

8 

16 

7 

3 

I 

Beam  Problems  in  One  and  Two  Unknowns  99 

9.  Observe  that  many  pairs  of  values  of  R  and  5  can  be 
found  that  satisfy  equation  (i),  alone,  and  equation  (3),  alone. 
Find  a  pair  that  satisfies  both  equations. 

10.  Find  as  in  problems  7,  8,  and  9  the  pair  of  values  of 
H  and  K  that  satisfies  both  of  the  equations: 

11.  Find  the  pair  of  values  of  M  and  L  that  satisfies  the 
equations 

M  —  2L=  3 
M-\-2L  =  ii. 

12.  A  weight  of  84  lb.,  hanging  on  a  stick  8  ft.  long  at 
a  point  2  ft.  from  the  middle,  is  raised  by  two  boys  who 
lift  at  the  ends  of  the  stick.     How  much  does  each  boy  lift? 

74.  From  the  foregoing  problems  it  follows  (i)  that  an 
equation  containing  two  unknowns  is  satisfied  by  many 
pairs  of  numbers,  (2)  that  two  different  equations  in  the  same 
two  unknowns  may  be  satisfied  by  a  single  pair  of  numbers. 

75.  The  single  pair  of  values  that  satisfies  the  conditions 
of  some  problems  can  be  found  from  two  equations  in  two 
unknowns  directly  as  follows. 

I.  The  sum  of  two  numbers  is  24^,  and  the  difference  is 
8  J.     What  are  the  numbers  ? 

Let  s  represent  the  smaller  number,  and  /  the  larger.  Then  by  the 
conditions  of  the  problem 

l  +  s  =  24i  (i) 


l-s=  8J.                                             (2) 

Adding  (i)  and  (2) 

2^  =  33                                                (3) 

i  =  i6J.                                             (4) 

Subtracting  (2)  from  (i) 

2S  =  l6                                                          (5) 

5=  8.                                               (6) 

Check:   The  sum  of 

i6i  and  8  is  24;   the  difference  (8  taken  from 

i6i)  is  8i. 

Therefore  the  smaller 

number  is  8  and  the  larger,  16 J. 

loo  Fir  St- Year  Mathematics 

2.  The  sum  of  two  numbers  is  24 J,  and  the  difference  is 
1 1  J.    What  are  the  numbers? 

3.  The  sum  of  two  numbers  is  12.  If  3  times  the  smaller 
is  subtracted  from  4  times  the  larger,  the  result  is  13.  Find 
the  numbers. 

Letting  /  represent  the  larger  number,  and  s  the  smaller, 

l  +  S  =  12  (i) 

To  find  /,  multiply  both  sides  of  equation  (i)  by  3,  and  add  the 
resulting  equation  to  equation  (2). 

To  find  s,  multiply  both  sides  of  equation  (i)  by  4,  and  subtract 
equation  (2)  from  the  resulting  equation. 

Check. 

4.  Solve  for  w  and  /: 

67^+4^=42 
5^-3^  =  16. 

76.  To  solve  beam  or  lever  problems  in  two  unknowns  it 
is  sometimes  convenient  to  use,  besides  the  law  of  leverages 
used  thus  far,  another  law  which  is  explained  in  the  following 
experiments. 

I.  Putting  a  weight,  x,  at  I3  (Fig. 
99),  and  an  equal  weight,  x,  at  x^,  it 
will  be  found  that  two  weights,  each 
equal  to  x,  hung  to  the  pan,  S,  will 
hold  the  apparatus  in  balance.  Begin- 
ning on  the  left,  record  the  relations 
for  balance  thus: 
Fig.  99  —x-\-2X—x=o. 

2.  With  two  weights,  each  equal  to  ic  at  I3,  and  two  weights, 
each  equal  to  x,  ^Xr^,  4  weights,  each  equal  to  x,  must  be  put 
at  S  for  balance.  But  3  or  5  weights,  x,  at  S  will  be  found  not 
to  balance.     Make  the  record  and  interpretation  thus: 


iiiliiniiiiiiiiiiiiniitinlliiii 


Bpam  Problems  in  One  and  Two  Unknowns 


lOl 


Record 

—  2X  +  ^X  —  2X=—X 

—  2X  +  ^X  —  2X=  +X 


Interpretation 
Balance 

Movement  downward 
Movement  upward. 


3.  If  2  weights,  X,  be  put  at  I3,  i  weight,  x,  at  Ij,  i  weight, 
X,  at  Tj,  and  2  weights,  x,  at  13,  6  weights,  each  equal  to  x,  at 
S  will  balance  the  apparatus,  but  neither  5  nor  7  weights,  x, 
will  balance  it.     The  results  will  run  thus: 


Record 
—2X—x-{-6x—x  —  2X=o 

—  2X—X  +  ^X  —  2X—X=  —X 

—  2X—X  +  'JX—X  —  2X=-^X 

4.  Write  the  equations  and  state  the  results  as  shown  by  a 
beam  for  each  loading  of  the  following  table: 


Interpretation 
Balance 

Bar  moves  downward 
Bar  moves  upward. 


No. 

14 

I3 

u 

ii 

s 

Tl 

Tj 

^3 

u 

I 

X 

X 

X 

X 

8^ 

X 

X 

X 

X 

II 

X 

X 

0 

0 

3^ 

0 

0 

X 

X 

III 

y 

0 

0 

y 

6y 

y 

0 

0    ■ 

y 

IV 

2y 

0 

zy 

0 

^sy 

Sy 

0 

2y 

0 
0 

V 

2y 

y 

zy 

y 

\?>y 

gy 

0 

sy 

5.  Use    the    same    bar,    supported    as    shown    in    Fig. 


100,  and  balance  it.  If  a 
weight,  X,  is  placed  at  both 
L  and  R,  two  weights,  each 
equal  to  3C  at  M  balance 
the  bar.  The  results  are: 
Record  Interpretation 
+x  —  2x+x=o    Balance. 

Try  the  same  weights 
at  J-.  and  R  and"33e  at  M; 
X  at  M. 


///////(///u//////////////y////////////////f// 


35 


7^ 


tU  -b  -U  .U « 


V  ^  tfc   \ 


Fig.  100 


I02 


First-Year  Mathematics 


6.  Eight  weights,  x,  at  Ij,  3  weights,  x,  at  R,  and  5  at  L, 
will  balance  the  bar.  If  the  8  weights,  x,  are  at  r,,  then  3 
weights,  X,  at  L  and  5  weights,  x,  at  R  are  needed  for  balance. 
The  results  are: 


Record 
+  $x—Sx+2X=o 
+^x—Sx  +  ^x=o 


Interpretation 
? 
? 


7.  Write  the  equations  and  state  whether  or  not  there  is 
balance  for  these  experiments: 


No. 

L 

I4 

13 

U 

li 

M 

Ti 

r. 

1-3 

r* 

R 

I 

6x 

0 

0 

Sx 

0 

0 

0 

0 

0 

0 

2X 

II 

Sx 

0 

0 

&x 

0 

AX 

0 

0 

0 

0 

4X 

III 

Sx 

X 

0 

6x 

0 

4X 

0 

0 

0 

X 

sx 

IV 

gx 

X 

0 

Sx 

0 

4X 

0 

0 

0 

X 

5x 

8.  When  a  bar  is  supported  in  two  places  (A  and  B,  Fig. 
100),  it  is  called  a  beam.  In  the  preceding  experiments  what 
is  the  test  as  to  whether  the  bar,  or  beam,  balances  ?  Observe 
that  in  all  these  experiments  the  forces  are  parallel  to  each 
other. 

77-  The  two  equations  used  in  the  following  problems  are 
derived  from  the  following  laws: 

Law  of  Forces. — The  algebraic  sum  of  all  the  forces  acting 
upon  the  material  object  {bar  or  beam)  must  equal  zero,  fr 
balance. 

Law  of  Turning -Tendencies  or  Leverages. — For  balance, 
the  algebraic  sum  of  all  the  turning-tendencies  must  equal 
zero. 


Beam  Problems  in  One  and  Two  Unknowns  103 

Problems  Applying  Two  Unknowns 

1.  A  box  12  ft.  long  of  a  three-horse  coal  wagon  is  loaded 
with  6  tons  of  coal.  If  the  box  extends  2  ft.  in  front  of  the  front 
axle  and  4  ft.  back  of  the  rear  axle,  what  are  the  weights  on  the 
front  and  rear  axles?     See  Fig.  loi. 

♦-Z-4* - -4" 4^ -2f-^  - -4'- - ♦  <._3'-X-_6'---.|--3*-i^ 6--* 

Fig.  ioi  Fig.  102 

2.  A  lumber  wagon  is  coupled  out  to  a  distance  of  9  ft. 
between  the  axles  and  loaded  with  a  pile  of  lumber  3^  ft.  X4  ft. 
X18  ft.  The  load  extends  3  ft.  in  front  of  the  front  axle  and 
the  material  averages  48  lb.  per  cubic  foot.  What  are  the 
pressures  on  the  axles  due  to  this  load?    Fig.  102. 

3.  A  wagon  box,  EFGO  (Fig.  103),  10  ft.  long  and  loaded 
with  40  bu.  of  wheat  weighing  60  lb.  per  bu.,  extends  i^  ft.  in 
front  of  the  front  axle,  B,  and  2^  ft.  behind  the  rear  axle,  A. 
What  is  the  load  on  each  axle  ? 

r-7---  --^- > 

1  I 

♦x    -no        -10  +5 

_  Fig.  104    ■ 

Fig.  103 

4.  Suppose  a  bar  10  ft.  long,  weighing  30  lb.,  is  used  by 
two  men,  one  grasping  it  at  each  end,  to  carry  a  load  of  170 
pounds.  How  many  pounds  must  each  man  carry,  if  the 
load  is  attached  2  ft.  from  the  left  end  ? 

^.  Measuring  lever-arms  from  the  middle  point  of  the  bar  show  that 
the  equations  are 

x  +  y  =  200  (i) 

-5a;-f-5io-o  +  5y=o.  (2) 

The  zero  term  in  (2)  arises  from  the  leverage  of  the  weight  of  the 

bar,  which  is  30*0,  zero  being  the  lever-arm.     But  30*0  =  0,  for  mani- 


I04  First-Year  Mathematics 

festly,  a  weight  hanging  at  the  middle  point  can  have  no  tendency  to 
turn  the  bar  around  this  point,  or,  what  amounts  to  the  same  thing, 
the  turning-tendency  about  this  point  equals  zero. 
Solve  the  equations. 

5.  A  foot-bridge  15  ft.  long  between  supports  (15  ft.  span) 
rests  on  timbers  at  L  and  R  (Fig.  105).     The  bridge  weighs 


Pit""! 

•  "Ao     -1000  ' 


Fig.  105 

1,000  pounds.  Two  men,  whose  combined  weight  is  450  lb., 
stand  just  over  A,  5  ft.  from  the  end  L.  Find  the  pressures  on 
the  supports  at  L  and  R,  using  M  as  turning-point. 

6.  A  bridge  20  ft.  long  weighs  2,400  lb.  and  supports  two 
loads;  one  of  600  lb.,  4  ft.  from  the  left  end,  and  the  other, 
800  lb.,  15  ft.  from  the  left.  What  are  the  loads  borne  by 
the  supports  ? 

7.  If,  with  the  bridge  of  problem  6,  four  loads  of  450  lb. 
each  are  placed,  one  2  ft.  from  the  left  support,  the  second 
6  ft.,  the  third  9  ft.,  and  the  fourth  16  ft.  from  the  left  end,  what 
are  the  upward  forces  exerted  against  the  ends  of  the  bridge 
by  the  supports? 

8.  A  wagon  standing  on  a  culvert,  A  B  (Fig.  106),  has  on 
the  front  axle  a  load  of  2,500  lb.,  and  on  the  rear  axle  3,000 

« jo: — > 


m — I       t 

I     ~uw      -mo 


Fig.  106 
pounds.  The  front  wheels  are  4  ft.  and  the  rear  wheels  10  ft. 
from  the  left  end  of  the  culvert.  If  the  cvdvert  is  20  ft.  long, 
and  weighs  2,000  lb.,  what  are  the  pressures  at  the  supports? 
9.  What  would  the  pressures  be  if  the  front  wheels  stood 
9  ft.  from  the  left  support  (wagon  coupled  to  6  ft.  between 
the  axles)? 


Beam  Problems  in  One  and  Two  Unknowns 


105 


10.  Two  men,  lifting  at  the  ends  of  a  stick  8  ft.  long,  raise 
a  certain  weight.  What  is  the  weight,  and  at  what  point  does 
it  hang,  if  one  man  Ufts  25  lb.,  and  the  other  75  pounds? 
(Use  M  as  turning-point.  Fig.  107.) 

z 


8- 

« — d — » 


^ — r 

+15        -*« 


IT 


:?. 


<&   0 


^ 


«---d  --, 


~ri — 

+  2/  -Z40 

Fig.  108 


^f 


Fig.  107 

11.  Three  boys  desire  to 
carry  a  12-ft.  log,  weighing 
240  pounds  (Fig.  108).  Two 
of  the  boys  lift  at  the  ends 
of  a  hand-spike  placed  crosswise  underneath  the  log  and  the 
third  boy  carries  the  rear  end  of  the  log.  Where  must  the 
hand-spike  be  placed  that  all  may  Hft  equally  ?  (Use  O  as 
turning-point.) 

12.  Solve  problem  11  using  A  as  turning-point.    Observe 
that  the  distance  AB  is  6—d. 


Exercise  XIV 

Find  the  values  of  the  unknown  numbers,  and  check: 

1.  s  —  i4-\-t=o 
s—t  +  2   =0 

2.  (4)(w)  +  (-6)(  +  3)  +  (-/)(  +  i)=o 

w  — II  +3/=o 

3.  (2i)(3*)  +  (i6)(-3)  +  (2)(3/)=o 

—  2^— ioH-6^=o 

4.  —2>{io—iv)-\-{—2){+gi)+4x=o 

—w  —  46-\-x=o 

5.  if-|-i6  — 3/=o 
5w  — 2^  — II  =0 

6.  -3i^-i4+3i>'=o 

x+y  =  i2 


io6  First-Year  Mathemahcs 

7-  \l-^\r-\-i=o 

4 

Summary 

1.  Beam  problems  in  which  one  number  is  unknown  may 
be  solved  by  using  the  following  law: 

Law  of  Turning-Tendencies.  For  balance,  the  algebraic 
sum  of  all  the  turning-tendencies  must  equal  zero. 

2.  Beam  problems  in  which  two  numbers  are  unknown 
may  be  solved  by  using  the  law  of  turning-tendencies,  and 
the  following  law: 

Law  of  Forces.  TJie  algebraic  sum  of  all  the  forces  must 
equal  zero,  for  balance. 

When  only  one  number  is  unknown  the  Law  of  Forces  may  be 
used  alone. 

3.  An  equation  containing  two  unknowns  is  satisfied  by 
many  pairs  of  numbers. 

Two  different  equations  in  the  same  two  unknowns  may 
be  satisfied  by  a  single  pair  of  numbers. 

4.  To  solve  a  problem  leading  to  two  equations  in  two 
unknowns,  a  third  equation  is  derived  which  contains  only 
one  of  the  unknowns. 

5.  The  product  of  two  factors  is  zero,  if  one  of  the  factors 
is  zero. 


CHAPTER  VI 
PROBLEMS  IN  PROPORTION  AND  SIMILARITY 
Drawing  to  Scale 
78.  Problems   in   finding   distances    may   be    solved   by 
drawings  made  to  scale. 

I.  A  man  starting  at  O,  Fig.  109,  walks  45  yd-  east  and 
then   60  yd.  north.    What   is  the  direct  distance  from   the 
A- 


y 


± 


/ 


Fig.  109 
stopping-point  to  the  starting-point,  if  i  cm.  on  the  drawing 
represents  10  yards? 

2.  Find  the  direct  distance  from  the  starting-point  to 
stopping-point  in  problem  i  from  a  drawing  in  which  i  cm. 
represents  15  yards. 

107 


io8  First-Year  Mathematics 

3.  A  man  walks  80  yd.  south,  then  144  yd.  east,  and  then 
120  yd.  north.  Find,  by  a  diagram,  his  distance  from  the 
starting-point,  letting  i  cm.  on  squared  paper  represent 
12  yards. 

4.  Two  men  start  from  the  same  point.  One  walks  5 
mi.  west,  and  then  3  mi.  north;  the  other  walks  4  mi.  south, 
and  then  5  mi.  east.     How  far  apart  are  they  ? 

5.  Draw  a  line  3  in.  long,  and  let  it  represent  a  distance 
of  48  feet.  What  distance  is  represented  by  i  inch  ?  By 
2  inches?  By  6  inches?  By  i\  inches?  By  2f  inches? 
By  i^V  inches? 

In  problem  5,  the  drawing  is  said  to  be  nuide  to  a  scale  of  i  inch 
to  16  feet. 

6.  Draw  to  the  same  scale:  8  feet;  12  feet;  24  feet;  28  feet. 

7.  If  a  line  5  in.  long  represents  a  distance  of  75  mi.,  what 
is  the  scale  ? 

8.  If  .7  of  an  inch  on  a  map  represents  a  distance  of.  21 
mi.  on  the  earth,  how  many  inches  represent  87  miles? 
What  scale  is  used  ? 

9.  Draw  a  plan  of  a  rectangular  field  16  rods  long  and  12 
rods  wide,  using  the  scale  of  i  in.  to  4  rods  (i  in.  =4  rods), 
and  find  the  distance  in  rods,  diagonally  across  the  field. 

Indicate  the  scale  on  all  scale-drawings. 

10.  Draw  to  the  scale,  ^  cm.  to  i 
ft.,  a  plan  of  a  room  24  ft.  by  18  ft., 
and  find  the  distance  diagonally  across 
the  floor. 

11.  Draw  to  the  scale,  2  cm.  to 
5  ft.,  a  plan  of  the  end  of  the  house  in 
Fig.  iio,  and  find  the  height  of  the 
top  of  the  roof  from  the  ground.  Use 
a  protractor  to  draw  the  angle  42°. 


Problems  in  Proportion  and  Similarity  109 

Ratio 

79.  The  ratio  of  6  to  3  is  f,  or  2;  of  3  to  4  is  f ;  of  a  to  6 

is  7 .    The  ratio  of  6  to  3  is  sometimes  written  6:3;  of  3  to  4, 

3  :  4;  and  of  a* to  b,  a  :  b. 

80.  The  ratio  of  any  number-  to  another  number  is  the 
quotient  found  by  dividing  the  first  number  by  the  second. 
Thus  §  is  the  ratio  of  2  to  3.  Any  fraction  may  be  regarded 
as  an  expression  of  the  ratio  of  the  numerator  to  the  denomi- 
nator. 

1.  Write  in  two  other  ways  the  following  ratios: 

(i)  5  to  20  (7)  a+b  to  c 

(2)  18  to  25  (8)  x+y  to  c-{-d 

(3)  25  to  18  (9)  a— &  to  c+d 

(4)  X  to  y  (10)  ax-\-ay  to  a 

(5)  c  to  d  (11)  3-'v  +  2  to  ab 

(6)  dtoc  (12)  a+b  to  x. 

2.  How  do  two  numbers  whose  ratio  is  i  compare  in  size  ? 

3.  What  is  the  ratio  of  the  cost  of  5  yd.  of  silk  at  $1 .  50 
to  that  of  50  yd.  of  cotton  at  12^  cents  ? 

4.  What  is  the  ratio  of  the  length  of  the  field  (problem  9, 
p.  108)  to  the  width  ?     Of  the  length  of  the  plan  to  the  width  ? 

5.  What  is  the  ratio  of  the  length  of  the  room  (problem  10, 
p.  108)  to  the  width  ?     Of  the  length  of  the  plan  to  the  width  ? 

6.  What  is  the  ratio  of  i  yd.  to  i  foot  ?  Of  i  yd.  to  i 
inch  ?  Of  I  yd.  to  6  inches  ?  Of  3  yd.  to  3  inches  ?  Of  3 
yd.  to  3  feet  ? 

7.  What  is  the  ratio  of  i  lb.  to  i  ounce  ?  Of  i  oz.  to  5 
pounds  ?     Of  I  ton  to  500  pounds  ?    Of  5  lb.  to  5  ounces  ? 

8.  What  is  the  ratio  of  i  mi.  to  i  yard?  Of  i  mi.  to  j 
foot  ?     Of  I  mi.  to  880  feet  ?     Of  i  mi.  to  880  miles  ? 


no  First-Year  Mathematics 

8 1.  Problems  6,  7,  and  8  show  that  magnitudes  must  be 
expressed  in  the  same  unit  before  the  ratio  can  be  expressed 
as  a  single  number. 

1.  Draw  three  triangles  in  each  of  which  the  angles  are 
respectively  35°,  65°,  and  80°.  Are  all  the  triangles  neces- 
sarily of  the  same  size  ?     Do  all  have  the  same  shape  ? 

2.  Draw  three  triangles  of  different  sizes  in  each  of  which 
the  angles  are  30°,  60°,  and  90°.  Compare  the  triangles  as  to 
shape.  Measure  all  sides  of  the  triangles.  Find  the  ratios 
of  the  sides  of  the  first  triangle  to  the  corresponding  sides  of 
the  second;   to  the  corresponding  sides  of  the  third  triangle. 

3.  Draw  triangles  of  diflferent  sizes  having  the  angles  52°, 
112^°,  and  15^°.  Compare  the  triangles  as  to  shape.  Find 
the  ratio  of  the  sides  of  one  of  the  triangles  to  the  corresponding 
sides  of  another. 

4.  Draw  three  triangles  each  of  which  has  angles  90°,  25°, 
and  65°.  Are  all  of  the  triangles  necessarily  of  the  same  size 
and  shape  ?     Compare  the  ratios  of  corresponding  sides. 

5.  Draw  a  triangle.  Draw  another  having  the  angles 
equal  respectively  to  the  angles  of  the  first  triangle.  Are  the 
two  triangles  necessarily  of  the  same  size  and  shape?  Com- 
pare the  ratios  of  the  corresponding  sides. 

Similar  Triangles 

82.  Triangles  having  the  same  shape  are  called  similar 
triangles.  Similar  triangles  are  not  necessarily  of  the  same 
size. 

I.  Draw  a  triangle  with  sides  4  in.  and  5  in.,  respectively, 
and  an  angle  of  50°  included  between  them.  First  draw  it 
actual  size  and  then  to  the  scale  of  f  in.  =  1  inch.  Are  the  tri- 
angles similar  ?  Why  ?  Measure  with  a  protractor  the  pairs 
of  corresponding  angles.  Find  the  ratios  of  the  corresponding 
sides. 


Problems  in  Proportion  and  SimUarity  n  i 

2.  Two  sides  of  a  triangle  are  2f  cm.  and  5.5  centimeters. 
The  included  angle  is  60°.  Two  sides  of  another  triangle 
are  4I  cm.  and  11  cm.,  and  the  included  angle  is  60°.  Draw 
the  triangles.  Are  they  similar?  Give  reason  for  the 
answer.  Measure  the  pairs  of  corresponding  angles  and  find 
the  ratios  of  the  corresponding  sides. 

3.  Draw  a  triangle.  Draw  another  with  sides  respectively 
double  the  lengths  of  the  first.  Compare  the  triangles  as  to 
shape.  Measure  the  corresponding  angles.  What  is  the  ratio 
of  the  corresponding  sides  ? 

4.  Two  sides  of  a  triangle  are  4.5  cm.  and  9.5  cm.;  the 
included  angle  is  70°.  Two  sides  of  another  triangle  are 
5f  cm.  and  13 J  cm.  and  the  included  angle  is  70°.  Are  the 
triangles  similar?  Compare  the  corresponding  angles.  Find 
the  ratios  of  the  corresponding  sides. 

83.  The  triangles  of  problems  i,  2,  3,  §82,  though  differing 
in  size,  have  the  same  shape  and  have  the  corresponding  angles 
equal.  Notice  also  that  each  side  of  the  smaller  triangle 
(problem  i)  is  |  of  the  length  of  the  corresponding  side  of  the 
larger  triangle. 

All  similar  triangles  may  be  regarded  as  the  same  triangle 
drawn  to  different  scales.  They  may  be  regarded  as  the  same 
triangle  magnified,  or  minified  to  a  definite  scale. 

1.  Draw  two  triangles  having  the  same  shape  but  different 
sizes.  Measure  two  corresponding  pairs  of  sides  in  each  and 
compare  the  ratios.  Measure  another  pair  of  corresponding 
sides  and  compare  the  ratios. 

2.  According  to  problems  i,  2,  3,  §82,  and  problem  i, 
§83,  what  seems  to  be  true  of  the  ratios  of  corresponding 
pairs  of  sides  of  triangles  having  the  same  shape  (similar 
triangles)  ? 


112 


First-Year  Mathematics 


84.  Two  triangles  are  similar  when  the  corresponding  angles 
are  equal  and  when  the  ratios  of  the  corresponding  sides  are 
equal. 

I.  In  the  two  similar  triangles  of  Fig.  iii,  (i),  if  a=4 
in.,  A  =12  in.,  and  6=4  in.,  how  long  is  J5? 


»V  V 


(Hi 


Fig.  Ill 


2.  In  the  similar  triangles  of  Fig.  11 1,  (2), 

If  a =4  in.,  .4  =  12  in.,  and  &  =  5  in.,  how  long  is  5? 
If  a=3  in.,  6=8  in.,  and  5  =  32  in.,  how  long  is  A? 
If  a=x  in.,  6=8  in.,  and  5  =  32  in.,  how  long  is  ^4  ? 

3.  In  the  similar  triangles  in  Fig.  11 1,  (3), 

If  A  =21  in.,  6=9  in.,  and  6  =  27  in.,  how  long  is  a? 
If  a  =  5i  in.,  ^4=22  in.,  and  ^  =  30  in.,  how  long  is  6? 

4.  In  the  similar  triangles  in  Fig.  iii,  (4), 

If  a =3  in.,  A=S  in.,  and  C  =  5  in.,  how  long  is  c? 
If  A  =24  in.,  c=4  in.,  and  C  =  7  in.,  how  long  is  a? 
If  A  =y  in.,  c=4  in.,  and  C  =  j  in.,  how  long  is  a? 

5.  In  Fig.  112,  if  the  stake  3  ft.  high  casts  a  shadow  8  ft. 


Problems  in  Proportion  and  Similarity 


"3 


long,  and  the  tree,  at  the  same  time,  casts  a  shadow  80  ft.  long, 
how  high  is  the  tree  ? 


6.  Are  triangles  O  A  B  and  O  H  K  (Fig.   113)   similar  ? 
Give  reason  for  answer.     How  long  is  :x;  ? 


Fig.  113 

7.  A  boy  holds  a  pencil,  A  B  (Fig.  114),  2  ft.  from  his  eye, 
so  that  it  covers  a  flag-pole  360  ft.  distant.  To  make  the 
triangles  E  A  B  and  E  F  K  similar,  how  must  the  pencil  be 
held  ?     If  the  pencil  is  6  in.  long,  how  high  is  the  pole  ? 


8.  A  lumberman  who  is  5  ft.  tall  wishes  to  find  a  tree  60 
ft.  to  the  first  limbs.     He  drives  a  stake  in  the  ground  and  places 


114 


First- Year  Mathematics 


his  feet  against  it  as  in  Fig.  115.  If  the  stake  is  4  ft.  high, 
how  far  must  it  be  placed  from  the  foot  of  the  tree,  that  he 
may  determine  whether  or  not  the  trunk  is  60  ft.  to  the  limbs  ? 


Fig.  115 

9.  The  gables  of  a  house  and  of  a  porch  have  the  same 
shape.  The'  sides  of  the  porch-gable  are  7  ft.,  7  ft.,  and 
10  feet.  The  longest  side  of  the  house-gable  is  25  feet.  What 
is  the  ratio  of  the  corresponding  sides?  How  long  are  the 
other  two  sides  of  the  house-gable  ? 

10.  The  sides  of  a  triangle  are  8,  10,  and  13.  The  shortest 
side  of  a  similar  triangle  is  11.  What  is  the  ratio  of  the  cor- 
responding sides  ?    Find  the  other  sides. 

11.  The  sides  of  a  triangle  are  4.6  cm.,  5.4  cm.,  and  6 
centimeters.  The  corresponding  sides  of  a  similar  triangle 
are  x  cm.,  y  cm.,  and  15  centimeters.     Find  x  and  y. 

12.  The  sides  of  a  triangle  are  1,2,  and  3,  and  the  longest 
side  of  a  similar  triangle  is  20.  Find  the  other  sides  of  the 
second  triangle. 

13.  Draw  a  triangle  (Fig.  116)  having  two  of  the  sides 
A  B  and  A  C  equal  to  10  in.  and  12  in.,  including  any  con- 


j^T----'.-  :::  ::::::iii-i 

:±±    "::  :::_  

—     z:i:,z'.'.~-'.z.z _ 

"Ti:"i""5"""!fs"i::"i"i 

— ^                        '.                    ^  s 

3'"'::E!!!! !'£::::: 

Fig.  116 


Problems  in  Proportion  and  Similarity  115 

venient  angle  between  them.  Call  the  third  side  the  base. 
Through  a  point  on  A  B,  5  in.  from  the  vertex  A,  draw  DE 
parallel  to  the  base.  Measure  the  distance  A  E.  How  does 
the  ratio  of  the  corresponding  parts  of  the  sides  A  B  and  A  C, 
cut  off  by  the  parallel  D  E,  compare  with  the  ratio  of  the  sides 
A  B  and  A  C  ? 

14.  Draw  a  parallel  to  the  base  (Fig.  116)  through  a  point 
of  the  lo-in.  side,  2^  in.  from  the  vertex,  and  measure  the 
distance  from  the  vertex  to  the  crossing-point  of  the  2 J  in. 
parallel  with  the  12-in.  side.  Compare  the  ratio  of  the  cor- 
responding parts  of  the  sides  with  the  ratio  of  the  sides  them- 
selves. 

15.  Compare  the  ratios  of  the  corresponding  parts  of  the 
sides  made  by  parallels  to  the  base,  through  a  point  of  the  10  in.- 
side  3  in.  from  the  vertex;  6  in.  from  the  vertex;  i\  in.  from 
the  vertex;   7 J  in.  from  the  vertex. 

85.  A  line  drawn  parallel  to  one  side  of  a  triangle  divides 
the  other  two  sides  into  corresponding  parts  having  the  same 
ratio  as  the  sides  themselves. 

Any  number  of  parallels  to  the  base  of  a  triangle  divide 
the  other  two  sides  into  parts  having  the  same  ratio. 

1.  In  Fig.  117,  AB  =  2i,  AC  =  3S, 
and  A  D  =3.  D  E  is  parallel  to  B  C. 
Find  the  value  of  x. 

2.  Show  that  triangles  A  D  E  and 
ABC,  Fig.  116,  are  similar. 

86.  If  a  line  is  drawn  parallel  to  ' 

one  side  of  a  given  triangle,   meeting  the  other  two  sides,  a 
/^r—.^^  triangle  is  formed  which  is  sim- 

/y^~^^I-^^^S~~~^^  ^0,^  to  the  given  triangle. 

{^P^- — i ^^^^^^^^^z»        I.  AB  (Fig.  118)  is  parallel 

^»"- ,____^__y> '^  to  C  D.     Find  C  D,  the  distance 

^         „  across  the  lake. 

Fig.  118 


ii6 


First-Year  Mathematics 


2.  Draw  a  triangle,  as  A  B  C,  Fig.  119.     Divide  A  B  and 
A  C  into  parts  having  the  ratio  i :  3.      Connect  the  points  of 


Fig.  119 

division,  D  and  E,  by  a  straight  line.  Divide  A  B  and  A  C 
in  other  ratios,  as  2:3,  1:1,  3:7,  1:9,  1:4,  and  connect  the 
corresponding  points  of  division  by  straight  lines.  Notice  that 
these  lines  are  parallel,  that  is,  they  will  not  meet,  however 
far  extended. 

87.  //  two  sides  of  a  triangle  are  divided  into  parts  having 
the  same  ratio,  the  line  joining  the  points  of  division  is  parallel 
to  the  third  side  of  the  triangle. 

Problems  in  Surveying 

88.  Problems  in  finding  directions  and  distances  may  be 
solved  by  drawings  made  to  scale. 

In  Fig.  121  the  direction  shown  by  the  arrow  is  read 
"30°  east  of  north;"  in  Fig.  122,  "50°  west  of  south;"  in 
Fig.  123,  "20°  east  of  south." 

89.  The  direction  of  a  line,  when  indicated  by  the  angle 
it  makes  with  the  north-south  line,  is  called  the  bearing  of 
the  line. 


Problems  in  Proportion  and  Similarity  117 

Surveyors  use  the  surveyors'  compass,  Fig.  128,  to  measure 
the  bearings  of  lines. 

jr 


V 


I 
I 


i 

Fig.  121 


T  1 


♦ 

»  a 

Fig.  122  Fig.  123 


r  .  r 


I  I  I 


I  I 

I  I 


Fig.  1 25  Fig.  i  26  Fig.  i  27 

1.  Read  the  bearings  of  the  arrows  in  Figs.  124,  125,  126, 
127. 

2.  With  a  ruler  and  protractor  draw  lines  having  the  fol- 
lowing bearings: 

65°  east  of  south  47^°  west  of  south 

65°  east  of  north  43°    west  of  north. 

90.  The  bearing  of  a  point  B  from  a  point  A  is  the  bear- 
ing of  the  line  A  B  with  reference  to  the  north-south  line 
through  A. 


ii8 


First-Year  Mathematics 


I.  In  Fig.  129,  read  the  bearings  of 

A  from  O  B  from  O  C  from  O 

O  from  A  O  from  B  O  from  C. 

2.  The  bearing  of  fort  A 
from  fort  B,  both  on  the  sea- 
coast,  is  65°  west  of  north. 
An  enemy's  vessel  at  anchor 
off  the  coast  is  observed  at  A 
to  bear  northeast;  at  B,  north- 
west.     The  forts  are  known 


_l 


Fig.  128 

to  be  7  mi.  apart.     Find  by  drawing  a  plan  (scale:  2  cm.  =  i 
mi.)  the  distance  from  each  fort  to  the  vessel. 

In  solving  problems  like  2,  draw  first  a  sketch  of  the  distances  and 
directions,  and  then  make  the  scale  drawing. 

3.  Find  the  distance  P  Q,  if  Q  is  6.4  mi.  east  and  9.8  mi. 
north  of  P  (scale:   i  in.  =  2  mi.).     What  angle 
does    P  Q    make   with    the    north-south    line 
through  P  ? 

4.  A  hill  in  a  battle-field  obstructs  the  view 
from  a  battery  at  B,  Fig.  130,  to  the  enemy's 
fort  at  F.  A  point  H  is  found  at  the  bottom 
of  the  hill,  from  which  F  is  observed  to-  bear        Fig.  130 


Problems  in  Proportion  and  Similarity 


119 


4  mi.  northeast.      If  H  is  6.25  mi.  northwest  of  B,  what  is 
the  distance  F  B,  and  the  bearing  of  F  from  B  ? 

The  triangle  in  the  drawing,  problem  4,  is  similar  to  the  triangle  in 
the  field,  for  the  first  triangle  is  the  same  as  the  second  with  all  sides 
minified  or  reduced  in  a  definite  ratio,  the  angles  remaining  unchanged. 

5.  A  tree  at  Q  is  6 . 5  rods  north  of  R,  and  9  rods  west  of  S. 
What  is  the  distance  and  bearing  of  S  from  R?  Show  that 
the  triangle  in  the  drawing  is  similar  to  the  triangle  in  the 
field. 

6.  A  man  wishes  to  measure  the  width  of  a  river  without 
crossing  it.     The  river  flows  due  west.     Standing  at  A,  on 


Fig.  131 


I2C 


First- Year  Mal/iematia 


the  bank,  he  observes  a  tree  on  the  other  bank  in  the  direction, 
20°  east  of  north.  He  walks  50  rods  east  along  the  bank  to 
B,  and  there  observes  the  tree  in  the  direction,  60°  west  of 
north.     Find  the  width  of  the  river. 

91.  In  the  preceding  problems  angles  have  been  measured 
by  referring  them  to  the  north-south  line.  By  means  of  the 
engineers'  transit  (Fig.  131)  angles  in  any  position  may  be 
measured. 

For  rough  measurements,  an  angle-measurer  can  be  constructed  by 
tacking  a  protractor  on  a  board  (Fig.  132).  A  ruler  with  a  pin  stuck 
in  it  at  each  end  can  be  used  for  sighting. 

Cr 


Mr<L' 


Fig.  132 


Fig.  133 


1.  Draw  a  plan  of  a  garden  plot  from  the  data  of  Fig.  133. 
Find  the  length,  in  rods,  of  B  C,  and  of  the  perpendicular 
from  B  to  A  C.     (Scale:   i  in.  =10  rods.) 

2.  Draw  triangles  having  the  following  parts: 

4  in.,  4  in.,  and  the  included  angle  60° 
3  in.,  4  in.,  and  the  included  angle  90° 
3  in.,  3^  in.,  and  the  included  angle  50°. 
Measure  the  third   sides   of  the  triangles.      Compare  your 
results  with  those  of  other  members 
of  the  class.     What  truth  about  tri- 
angles do  you  infer  ? 

3.  A  railroad  surveyor  wishes  to 
measure  across  the  swamp  A  B,  Fig. 
1 34.  He  measures  the  distance  from 
a  tree  at  A  to  a  stone  at  C  and  finds 


Problems  in  Proportion  and  Similarity 


121 


it  to  be  165  feet.  The  distance  from  a  tree  at  B  to  the  stone 
is  150  feet.  Find  the  distance  in  feet  across  the  swamp,  the 
angle  at  C  being  80°  (scale:  i  cm.  =  15  ft.).  Show  that  the  tri- 
angle in  the  drawing  is  similar  to  the  surveyed  triangle. 

4.  To  measure  the  width,  A  C, 
of  a  stream  (Fig.  135),  without 
crossing  it,  an  engineer  lays  off  a 
line,  B  C,  on  one  side  of  the  river, 
and  measures  (with  a  transit)  the 
angles  at  B  and  C.     Draw  a  triangle  Fig.  135 

to  scale  from  the  data  in  the  figure,  and  determine  the  width  of 
the  river. 

5.  Draw  triangles  from  the  following  data: 

A  B  =  3J  in.,  angle  A  =  15°,  angle  B  =  i3o° 
A  B=3i  in.,  angle  B=9o°,  angle  C=  60° 
A  B=3^  in.,  angle  0  =  55°,  angle  A=  20°. 

In  each  triangle  measure  the  sides  not  given  and  compare  as 
in  problem  2.     What  do  you  infer? 

6.  A  boy  wishes  to  determine  the  height,  H  K  (Fig.  136), 
of  a  factory  chimney.  He  places  the  angle-measurer  first  at 
B  and  then  at  A  and  measures  the  angles  x  and  y.     The 


Fig.  136 

angle-measurer  lies  on  a  box,  or  tripod,  3^  ft.  from  the 
ground.  A  and  B  are  two  points  in  line  with  the  chimney  and 
50  ft.  apart.  What  is  the  height  of  the  chimney  if  the  ground 
is  level  and  if  x=6^°  and  ^'=33^°  ? 


122  First- Year  Mathematics 

7.  Classify  triangle  ACK,  Fig.  136,  as  to  its  angles. 
Triangle  BCK. 

8.  How  many  degrees  are  there  in  each  angle  at  K,  Fig. 
136  ?    In  the  angle  adjacent  to  angle  x  ? 

92.  A  telescope  is  pointed  hori- 
zontally toward  a  tower  (Fig.  137), 
and  the  farther  end  is  then  raised 
(elevated)  until  the  telescope  points  to 
the  top  of  the  tower.  The  angle 
Fig.  137  through  which  the  telescope  turned  is 

the  angle  of  elevation  of  the  top  of  the  tower,  from  the  point 
of  observation. 

1.  From  point  A,  Fig.  136,  what  is  the  angle  of  elevation 
of  the  top  of  the  chimney  ?     From  point  B  ? 

2.  When  the  angle  of  elevation  of  the  sun  is  25°,  a  building 
casts  a  shadow  90  ft.  long,  on  level  ground.  Find  the  height 
of  the  building. 

3.  Find  the  angle  of  elevation  of  the  sun  when  a  tree 
40  ft.  high  casts,  on  level  ground,  a  shadow  60  ft.  long. 

4.  On  the  top  of  a  tower  stands  a  flagstaff.  At  a  point, 
A,  on  level  ground,  50  ft.  from  the  base  of  the  tower,  the 
angle  of  elevation  of  the  top  of  the  flagstaff  is  35°.  At  the 
same  point  A,  the  angle  of  elevation  of  the  top  of  the  tower  is 
20°.     Find  the  length  of  the  flagstaff. 

93.  A  telescope  at  T,  on  „     i.„-^-ui  ii« 


>!.>■ 


the  top  of  a  cliff  (Fig.  138), 

is  pointed  horizontally,   and  ♦»^ 

then  the  farther  end  is  lowered  ^  ^  " 

(depressed)  until  the  telescope  5*^ 

points  to  the  buoy  at  B.     The  ^^^-  ^38 

angle  through  which  the  telescope  turned  is  the  angle   of 

depression  of  the  buoy  from  the  point  T. 


Problems  in  Proportion  and  Similarity  123 

1.  If  the  height  of  the  cliff,  Fig.  138,  is  100  ft.,  and  the 
angle  of  depression  of  the  buoy,  as  seen  from  T,  is  40°,  what 
is  the  distance  of  the  buoy  from  the  bottom  of  the  cliff  ? 

2.  A  boat  passes  a  tower  on  which  is  a  search-light  120  ft. 
above  sea-level.  Find  the  angle  through  which  the  beam  of 
light  must  be  depressed,  from  the  horizontal,  so  that  it  may 
shine  directly  on  the  boat  when  it  is  400  ft.  from  the  base  of 
the  tower. 

3.  From  the  top  of  a  cliff  150  ft.  high,  the  angle  of  depres- 
sion of  a  boat  is  25°.  How  far  is  the  boat  from  the  top  of  the 
cliff? 

4.  From  a  lighthouse,  situated  on  a  rock,  the  angle  of 
depression  of  a  ship  is  12°,  and  from  the  top  of  the  rock,  it  is 
8°.  The  height  of  the  Ughthouse  above  the  rock  is  45  feet. 
Find  the  distance  of  the  ship  from  the  rock. 

Proportion 
94.  Comparing  the  ratios  of  the  areas  of  figures  with  the 
ratios  of  corresponding  dimensions  leads  to  proportion. 

1.  Two   rectangles,   as    ABCD    and    * , 

' — ' — ' — I 
EFGH  (Fig.  139),  have  equal  altitudes,  h.    J    j    ;    j 

The  bases  are  7  in.  and  4  in.  respectively. 

What  are  the  areas?     Find  the  ratio  of 

the  areas.     Find  the  ratio  of  the  bases. 

How  does  the  ratio  of  the  areas  compare  Fig.  139 

with  the  ratio  of  the  bases  ? 

2.  The  altitude  of  a  rectangle  is  10  in.  and  the  base  is 
4  feet.  The  altitude  of  another  rectangle  is  20  in.  and  the 
base  is  4  feet.  What  is  the  ratio  of  the  areas  ?  Of  the  alti- 
tudes ?     Compare  the  ratios. 

3.  Two  rectangles  have  bases  20  ft.  and  25  ft.,  and  an 
altitude  of  15  feet.  Express  by  an  equation  that  the  ratio  of 
the  areas  equals  the  ratio  of  the  bases. 


124 


First-Year  Matliematics 


4.  The  dimensions  of  one  rectangle  are  a  and  h,  and  of 
another  a  and  c.  Compare  the  ratio  of  the  areas  with  the 
ratio  of  the  unequal  dimensions,  and  express  the  result  by  an 
equation. 

5.  How  does  the  ratio  of  the  areas  of  rectangles  having 
equal  bases  compare  with  the  ratio  of  the  altitudes?  How 
does  the  ratio  of  areas  of  rectangles  having  equal  altitudes 
compare  with  the  ratio  of  the  bases  ? 

6.  The  area  of  rectangle  ABCD  (Fig.  140),  is  80  sq. 
ft.,  and  the  base  is  10  yards.  What  is  the  area  x  of  rectangle 
EFGH,  having  the  same  altitude  and  a  base  equal  to  24 
yards  ? 


80 


-u 

Fig.  140  Fig.  141 

7.  Express  by  an  equation  the  relation  between  the  areas 
and  bases  of  the  rectangles  in  Fig.  141.     Find  the  base  x. 

K      X 

8.  The  equation  -  =  q  expresses  the  relation  between  the 

4     o 

ratio  of  the  areas  and  the  ratio  of  the  altitudes  of  two  rec- 
tangles.    Find  the  altitude  x. 

9.  Triangles  ABC  and  ADC  (Fig. 
142),  with  the  same  base,  have  altitudes  as 
shown.  What  is  the  ratio  of  the  areas  ?  Of 
the  altitudes  ?    How  do  the  ratios  compare  ? 

10.  If  two  or  more  triangles  have  equal 
bases,  how  does  the  ratio  of  the  areas  of  any  two  compare 
with  the  ratio  of  the  altitudes?  ^.        /lV 

II.  Two    triangles    (Fig.   143)    have   ^  t,  '        ^  '  ;  ^ 
equal  bases,  h,  and  altitudes  as  shown.  fig.  143 

The  ratio  of  the  areas  is  f .     Express  by  an  equation  the 


Problems  in  Proportion  and  Similarity 


125 


relation  between  the  ratio  of  the  areas  and  the  ratio  of  the 
altitudes.     Find  the  altitude,  x,  by  solving  the  equation. 

12.  Compare  the  ratios  of  areas  and  of  bases  of  triangles 
with  equal  altitudes  (Fig.  144).  Express  the  result  by  an 
equation. 


Fig.  144 


lU 


For 


and  T—-,  are  all    called    proportions, 


Fig.  145 

13.  Compare  the  ratios  of  areas  and  of  bases  of  the  parallelo- 
grams in  Fig.  145.     Express  the  result  by  an  equation. 

95.  An  equation  of  two  ratios  is  called  a  proportion. 

,     4    2    a    ac 
example, -=-,^=^, 

and  are  sometimes  written  thus:  4:6  =  2:3,  a:h=ac:hc,  and 
a:b=c:d.  The  last  maybe  read  "a  is  to  6  as  c  is  to  d." 
Read  the  other  two.  Numbers  that  form  a  proportion  are 
said  to  be  proportional. 

Four  lines  are  said  to  be  proportional  if  their  lengths  are 
proportional. 


Fig.  146 

1.  In  Fig.  146,  the  letters  a  and  b  denote  the  same  num- 
bers throughout.  How  do  the  areas  of  triangles  I,  II,  and 
III  compare  ?  I  and  IV  ?  II  and  VI  ?  Ill  and  V  ?  IV  and 
VIII  ?  II  and  IX  ?  IX  and  X  ?  VII  and  X  ?  Ill  and  X  ? 


X26  First- Year  Mathemaiics 

2.  Show  that  (i)  areas  of  rectangles  are  proportional  to  the 
bases,  if  the  altitudes  are  equal,  i.  e.,  the  ratio  of  the  areas 
equals  the  ratio  of  the  bases; 

(2)  areas  of  rectangles  are  proportional  to  the  altitudes,  if 
the  bases  are  equal; 

(3)  areas  of  rectangles  are  proportional  to  the  products  of 
the  bases  and  altitudes; 

(4)  areas  of  triangles  are  proportional  to  the  products  of  the 
bases  and  altitudes. 

3.  The  altitude  and  base  of  a  triangle  are  4  ft.  and  15  ft.  re- 
spectively. What  are  the  dimensions  ot  other  triangles  of  ^  the 
area  ?    Of  J  the  area  ?     Of  ^  the  area  ? 

4.  The  altitude  and  base  of  a  rectangle  are  6  in.  and  8  in. 
respectively.  What  are  the  altitudes  and  bases  of  triangles 
whose  areas  are: 

(i)  equal  to  the  area  of  the  rectangle  ? 

(2)  twice  the  area  of  the  rectangle  ? 

(3)  one-half  the  area  of  the  rectangle  ? 

5.  Answer  the  same  questions  when  the  altitude  and  base 
of  the  rectangle  are  a  and  b  inches  respectively. 

6.  A  triangle  and  a  rectangle  have  equal  bases  and  are 
equal  in  area.     How  do  the  altitudes  compare  ? 

7.  The  dimensions  of  a  rectangular  block  are  4  ft. X 15  ft. 
X 25  feet.  What  are  the  dimensions  of  other  rectangular 
blocks  having  §  the  volume  ?  J  the  volume  ?  ^  the  volume  ? 
J  the  volume  ?    -^  the  volume  ? 

8.  Two  rectangular  flower  beds  have  the  same  shape,  but 
are  different  in  size.  One  is  3  ft.  wide  and  5  ft.  long;  the 
other  is  12  ft.  wide.  How  long  is  it?  What  is  the  ratio  of 
the  corresponding  sides  ? 

9.  Two  books  have  the  same  shape.  One  is  5J  in.  wide 
and  7  J  in.  long.    The  other  is  15  in.  long.    How  wide  is  it  ? 


\ 


Problems  in  Proportion  and  Similarity  127 

10.  The  top  of  a  desk  and  a  rectangular  sheet  of  paper, 
12  in.  by  18  in.,  have  the  same  shape.  The  desk  is  2  ft. 
wide.     How  long  is  it  ? 

11.  A  city  block  and  a  lot  within  the  block  have  the  same 
shape.  The  lot  is  100  ft.  by  150  ft.  and  the  block  is  300  ft. 
wide.     How  long  is  it  ? 

12.  Is  2:5  =  5:15  a  proportion?  Is  2:7=8:25  a  propor- 
tion ?     Give  reasons  for  your  answers. 

96.  The  first  and  last  terms  of  a  proportion  are  called  the 
extremes ;  the  second  and  third,  the  means. 

1.  Compare  the  product  of  the  extremes  with  the  product 
of  the  means  in  the  proportion  2:5=6:15;  in  3:7=6:14;  in 
20:2=10:1;  in  12:3=4:1.  What  do  you  find  true  of  the 
products  ? 

2.  For  which  of  the  following  expressions  does  the  product 
of  the  first  and  last  number  equal  the  product  of  the  other  two  ? 

(i)  1:3=  4:12  (6)  2:3  =  20:3(1 

(2)  3:4=  6:12  (7)  8:80  =  3:33 

(3)  2:3=  8:ix  (8)  x:y=4x:4y 

(4)  5:6  =  10:12  •  (9)  ^a:2X=6a:6x 

(5)  8:3  =  15:3  (10)  x:sx  =  i:s. 

Is  there  any  proportion  in  this  list  in  which  the  product  of 
the  means  does  not  equal  the  product  of  the  extremes  ? 

Is  there  in  the  list  any  expression  that  is  not  a  proportion, 
for  which  the  product  of  the  first  and  fourth  number  is  equal 
to  the  product  of  the  second  and  third  ? 

97.  In  a  proportion,  the  product  of  the  means  equals  the 
product  of  the  extremes. 

This  is  a  convenient  test  of  proportionality, 
I.  By  this  test  tell  what  expressions  in  problem  2,  §96, 
are  proportions. 


128 


First-Year  Mathematics 


2.  If  7  =  J,   prove  that  a(i  =  6c. 

0    d 

3.  Divide  85  into  two  parts  in  the  ratio  2:3, 

4.  Divide  84  into  three  parts  proportional  to  3:4:5- 

5.  What  number  added  to  12  and  subtracted  from  30,  gives 
results  that  are  to  each  other  as  5:10? 

6.  Two  numbers  are  in  the  ratio  of  5:6.  If  12  is  sub- 
tracted from  each,  the  differences  are  in  the  ratio  3:4.  What 
are  the  numbers  ? 

7.  The  ratio  of  two  lines  is  2^:3!.  The  longer  line  is 
30  centimeters.     Find  the  shorter  line. 


I                    -           -   - 

i^i^I-   .- 

tV^i'i        .-       -  - 

g  J      ■=  c 

"    7     ^       ^^^ 

-   -/       \         -    i 

t ^ ^;» 

Fig.  147 

8.  In  Fig.  147  the  ratio  of  the  parts  of  AC  equals  the 
ratio  of  AB  to  BC.  AD  is  2  in.  less  than  DC.  Find  the 
lengths  of  AD  and  DC. 

9.  On  squared  .paper  make  an  accurate  drawing  for 
problem  8.  Measure  the  angles  at  B.  How  do  they  compare 
in  size  ? 

10.  If  6°  be  taken  from  one  of  two  complementary  angles 
and  added  to  the  other,  the  ratio  of  the  two  angles,  thus  formed, 
is  2:7.     Find  the  angles. 

11.  The  ratio  of  2  times  one  of  two  supplementary  angles 
to  8  times  the  other  is  1:2.     Find  the  supplementary  angles. 

12.  Three  angles  just  covering  the  plane  around  a  point 
are  to  each  other  as  2:3:4.     Find  them. 

13.  The  angles  of  a  triangle  are  as  1:2:3.     Find  them. 


Problems  in  Proportion  and  Similarity  129 

14.  The  acute  angles  of  a  right  triangle  are  as  2:5.     Find 
them. 

15.  Find  the  value  of  x  in  the  following  proportions: 

b)  i-H-  (8)  "-'     "-' 


X  •^  —  13     x  —  14 

(3)  -ii%=x:3  /  X  x  +  2^x+_i 

X  x+:^   x+i 

(4)  ^  =  rM7  (^^)  X- 64 =400:1 

szf^^^z:^  (11)  -■ 

x+2    4+x  3 

(6)  ^Z21=?  (,,)  ^  +  5_   x6 


(5) 


1-3^    I  4       ^  +  5 

98.  Proportions  may  be  written  from  equations  that 
express  the  equality  of  products. 

1.  The  statements  below  are  different  arrangements  of  the 
four  factors  in  the  equation 

8-7  =  14.4.  (i) 

A{)ply  the  test  of  proportionality  and  point  out  which 
statements  are  proportions: 

(2)  8:14=4:7  (6)  8:7  =  14:4 

(3)  l=V  (7)  tV=I 

(4)  4:7=8:14  (8)  8:4  =  7:14 

(5)  V-=l  (9)  A=J- 

2.  From  what  place  in  the  given  equation  (i)  were  the 
means  taken  in  the  proportions  ?    The  extremes  ? 

3.  Is  the  same  thing  true  of  any  of  the  statements  that  are 
not  proportions  ? 

4.  Try  to  write  a  false  expression  of  proportion  by  making 
both  factors  of  either  side  of  equation  (i)  the  means  of  the 
expression. 


130  First-Year  Mathematics 

5.  Try  to  write  a  correct  statement  of  proportion  using  any 
one  of  the  four  factors  of  the  given  equation  for  a  mean  and 
the  other  factor  on  the  same  side  for  an  extreme. 

6.  Write  four  proportions  from  3  •  28=4  •  21,  and  apply  the 
test  of  proportionality. 

7.  Write  four  proportions  from  a  •  i2&= 3a  •  4&,  and  test. 

99.  1/  the  product  of  two  numbers  equals  the  product  of  two 
other  numbers,  either  pair  may  be  made  the  means  and  the 
other  pair  the  extremes  of  a  proportion. 

I.  From  each  of  the  following  equations  write  at  least  four 
proportions  and  show  that  they  satisfy  the  test  of  propor- 
tionality: 

(i)  sc-iod=c-sod  .   ^    4_£^     2 

(2)  6 -210=18 -70        ^^^  '     's      5      ^ 

(3)  15  •7^  =  105-^         (6)  a-b=ab'i. 

(4)  a  '  bc=ab  '  c 

100.  Proportions  may  be  written  from  other  proportions. 

1.  Using  the  numbers  of  the  proportion  4:7  =  12:21, 
write  another  expression  in  proportion  form: 

(i)  without  changing  the  positions  of  the  4  and  the  21 
(the  extremes), 

(2)  without  changing  the  positions  of  the  7  and  the  12 
(the  means). 

2.  Interchange  the  means  only  in  5:7=15:21,  and  test  for 
proportionality.     Interchange  the  extremes  and  test. 

3.  If  a:  b=c:  d,  is  a:  c=b:  d?    Reason  for  answer. 

4.  If  a:  b=c:  d,  is  d:  b=c:  a?    Reason  for  answer. 

1 01.  If  the  means  or  the  extremes  of  a  proportion  are 
interchanged  the  resulting  expression  is  a  proportion. 

When  a  second  proportion  is  made  from  a  given  propor- 
tion by  interchanging  the  means,  or  by  interchanging  the 


Problems  in  Proportion  and  Similarity 


131 


extremes,  the  second  proportion  is  said  to  be  obtained  from 
the  given  proportion  by  alternation. 

A 


I.  In  Fig.  148,^=^ 


ABAC 
AE"AD 

Show  that     -r?^=-r^i=: 


AC    AD'  ^         o 

Fig.  148 

2.  In  two  equilateral  polygons  having  the  same  number  of 
sides,  the  corresponding  sides  are  proportional. 


that 


Then 


AB  A  B 

Proof:  Since  ^57^=1  (Fig.  149),  and  =^^^^  =  1  (Why?),  it  follows 


In  a  similar  way  prove 
BC 


ABA.Bi 
BC"BxCi 

AB  ^  BC 
AxB,~BxCx 

CD        DE 


Why? 
Why? 


BxC     C,Dx     DxE,     

3.  If  two  parallel  lines  are  intersected 
by  three  or  more  parallel  lines,  the  ratio 
of  any  two  parts  of  one  of  the  first  two 
parallels  equals  the  ratio  of  the  corre- 
sponding parts  of  the  other. 


Suggestion  for  the  proof: 

AB        BC 
AxBx~BxCx' 


Why? 


Fig.  150 


132 


First-  Year  Mathematics 


Fig.  151 


Variation 

1.  Let  the  vertical  side  of  a  square  (Fig.  151)  represent 
I  mile  and  the  horizontal  side  i  hour.     Graph  the  distance 

passed  over  in  i  hour,  2  hours,  3  hours, 

10  hours,  by  a  man  walking  at  the  rate  of  2  miles 
an  hour. 

Draw  the  lines  O  A,,  AiBj,  BiCj 

What  kind  of  a  line  is  O  Ei  ? 

How  do  the  angles  of  triangles  O  A  A,,  O 
B  Bi,  O  C  Ci,  etc.,  compare  in  size? 

What  is  the  ratio  of  A  A,  to  O  A  ?  of  B  B,  to 
O  B  ?  of  C  C,  to  O  C  ? 

2.  Show  that  the  triangles  O  A  A.,  O  B  B„ ,  etc., 

are  similar. 

What  is  the  ratio  of  any  distance-line  to  the  corresponding 
time-line  ? 

How  does  the  distance  change  when  the  time  is  doubled, 
trebled,  quadrupled,  etc.  ? 

How  does  the  distance-line  change  (vary)  when  the  time- 
line changes  ? 

3.  Write  the  equation  for  the  time  t  and  the  distance  d 
passed  over  by  a  man  walking  at  the  rate  of  2  miles  an  hour. 

What  are  the  values  of  rf  as  /  takes  the  values  1,2, 10? 

What  is  the  ratio  of  a  value  of  d  to  the  corresponding  value 
of/? 

4.  On  squared  paper  let  the  vertical  side  of  a  small  square 
denote  i  cent  and  the  horizontal  side  i  pound.     Graph  the 

cost  of  flour  at  3^  cents  per  pound,  for  i  lb.,  2  lb.,  3  lb., 

8  pounds. 

How  does  the  graph  show  the  cost  to  change  (vary)  as  the 
weight  is  doubled,  trebled,  etc.  ? 

How  does  the  graph  show  the  cost  to  vary  as  the  weight  is 
varied  ? 

What  is  the  ratio  of  the  cost  to  the  weight  ? 


Problems  in  Proportion  and  Similarity  133 

5,  Write  the  equation  for  the  weight  w  in  pounds,  and  the 
cost  c  of  flour  at  3^  cents  a  pound.  Find  the  values  of  c 
as  w  takes  the  integral  values  from  i  to  10. 

What  is  the  ratio  of  the  values  of  w  to  the  corresponding 
values  of  c  ? 

6.  Graph  the  areas  of  rectangles  with  the  altitude  5  and 

bases  i,  2,  3, 6.     How  does  the  area  vary  as  the  base 

varies  ? 

Write  the  equation  for  the  area  A  and  the  base  b  of 
rectangles  with  altitude  5. 

Does  a  change  in  value  of  b  change  the  value  of  ^  ? 

A 

Does  it  change  the  value  of  the  ratio   —  ? 

102.  The  number  x  is  said  to  vary  directly  as  the  number 
y,  if  the  ratio,  x:y,  remains  constant  (i.  e.,  does  not  change). 

1.  Graph  the  areas  of  triangles  with  altitude  6  and  bases 

I,  2,  3, 7.     Show  that  the  area  .4  of  a  triangle  varies 

directly  as  the  base  b. 

2.  Write  the  equation  for  the  area  ^  of  a  triangle  having 
the  altitude  h  and  the  base  b.  Show  from  the  equation  that 
the  area  A  varies  directly  as  the  base  b,  when  the  altitude  is 
constant. 

3.  Write  the  equation  for  the  area  ^  of  a  rectangle  having 
the  altitude  a  and  the  base  b.  Show  that  the  area  A  varies 
directly  as  the  base  b,  when  the  altitude  is  constant. 

4.  Cut  (from  card-board)  circles  with  diameters  of  various 
lengths.  With  a  string  or  thin  wire  measure  the  circumferences 
of  the  circles.  Find  the  ratio  of  the  circumference  of  each 
circle  to  the  diameter.     How  do  the  ratios  compare  ? 

5.  The  area  of  a  rectangle  varies  directly  as  the  base  if  the 
altitude  remains  constant;  and  when  the  area  is  27,  the  base 
is  3.    What  is  the  constant  ratio  of  the  area  to  the  base  ? 


134  First-Year  Mathematics 

What  is  the  equation  connecting  the  area  A  and  the  base  b  ? 
When  the  area  is  54,  what  is  the  base  ? 

6.  The  circumference  of  a  circle  varies  directly  as  the 
diameter.  The  constant  ratio  of  the  circumference  to  the 
diameter  is  3.14,  approximately.  Write  the  equation  for  the 
circumference  c  and  the  diameter  d.  When  the  circumference 
is  157,  what  is  the  diameter? 

7.  The  distance  d  through  which  a  body  falls  from  rest 
varies  directly  as  the  square  of  the  time  t  in  which  it  falls; 
and  a  body  is  observed  to  fall  400  ft.  in  5  seconds.  What  is 
the  constant  ratio  of  dtot'? 

Write  the  equation  for  d  and  /. 

How  far  does  a  body  fall  in  i  second  ?  In  2  seconds  ? 
In  3  seconds? 

8.  X  varies  directly  as  y,  and  when  x  =  2o,  y=4.  Find  the 
value  of  X  when  3*  =  17. 

9.  If  z  varies  as  x,  and  2=48  when  x=4',  find  z  when 
a;=ii. 

10.  The  turning- tendency  caused  by  a  weight  moved  along 
a  bar,  or  lever,  varies  directly  as  the  lever-arm.  The  turning- 
tendency  is  20  when  the  arm  is  5.  Find  the  turning- tendency 
when  the  arm  is  7. 

11.  Rectangles  with  area  6  sq.  ft.  have  bases  of  i,  2,  3, 

6.     For  each  rectangle  write  the  ratio  of  the  base  to 

the  reciprocal  of  the  altitude.     How  do  the  ratios  compare  in 
value  ? 

103.  The  number  b  is  said  to  vary  inversely  as  the  num- 
ber a,  if  the  ratio  of  b  to  the  reciprocal. of  a,  is  constant. 

I.  Rectangles  have  an  area  equal  to  10.  The  base  varies 
inversely  as  the  altitude.  What  is  the  constant  ratio  of  the 
base  to  the  reciprocal  of  the  altitude?  Write  the  equation 
for  the  base  b  and  the  altitude  a. 


Problems  in  Proportion  and  Similarity  135 

2.  y  varies  inversely  as  :».  y  =  2  when  x  =  i.  Find  the 
constant  ratio  of  y  to  the  reciprocal  of  x.  Write  the  equation 
for  X  and  y.     Find  y  when  x=%. 

3.  When  gas  in  a  cylinder  is  exposed  1  o  p  assure,  the  volume 
is  reduced  as  the  pressure  is  increased.  It  is  found  in  physics 
that  the  volume  varies  inversely  as  the  pressure. 

The  volume  of  a  gas  is  4  cubic  cm.,  when  the  pressure  is 
3  pounds.    What  is  the  volume  under  a  pressure  of  6  pounds  ? 

4.  The  speed  of  a  falling  body  varies  directly  as  the  time. 
Write  the  equation  for  the  speed  v  and  the  time  /. 

A  body,  falling  from  rest,  moves  at  the  rate  of  160  ft.  a 
second  5  sec.  after  it  began  to  fall.  What  will  be  the  speed 
attained  in  8  seconds  ? 

5.  The  distance  passed  over  by  a  body,  moving  at  a  con- 
stant rate,  varies  directly  as  the  time. 

Find  the  rate  of  a  train  which  travels,  at  uniform  rate, 
the  distance  of  225  mi.  in  6  hours. 

6.  A  stone  fell  from  a  building  560  ft.  high.  In  how 
many  seconds  did  it  reach  the  ground?     (See  7,  §  102.) 

7.  The  number  of  men  doing  a  piece  of  work  varies  inversely 
as  the  time. 

Twelve  men  can  do  a  piece  of  work  in  28  days.  In  how 
many  days  can  3  men  do  the  same  ? 

8.  The  time  /  of  oscillation  of  a  pendulum  varies  directly 
as  the  square  root  of  the  length  /.  A  pendulum  39.2  in. 
long  makes  one  oscillation  in  one  second.  Find  the  length 
of  a  pendulum  which  makes  an  oscillation  in  two  seconds. 


136  First-Year  Mathematics 

Summary 

1.  The  ratio  of  two  numbers  is  their  quotient. 

2.  Similar  triangles  are  triangles  that  have  the  same  shape. 
Similar  triangles  are  not  necessarily  of  the  same  size. 

3.  All  similar  triangles  may  be  regarded  as  the  same  tri- 
angle drawn  to  different  scales.  They  may  be  regarded  as 
the  same  triangle  magnified  or  minified  to  a  definite  scale. 

4.  Two  triangles  are  similar  when  the  corresponding  angles 
are  equal  and  when  the  ratios  of  tJie  corresponding  sides  are 
equal. 

5.  When  surveying  problems  are  solved  by  drawing  tri- 
angles to  scale,  the  triangle  in  the  drawing  is  similar  to  the 
surveyed  triangle. 

6.  A  proportion  is  an  equality  of  ratios. 

7.  Areas  of  rectangles  (and  triangles)  have  the  same  ratio 
(i)  as  the  bases,  if  the  altitudes  are  equal, 

(2)  as  the  altitudes,  if  the  bases  are  equal. 

8.  Areas  of  rectangles  (and  triangles)  have  the  same  ratio 
as  the  products  of  the  bases  and  altitudes. 

9.  The  extremes  of  a  proportion  are  the  first  and  last  terms; 
the  means  are  the  second  and  third  terms. 

10.  In  any  proportion  tJie  product  of  tJie  extremes  is  equal 
to  the  product  of  the  means.  This  is  a  convenient  test  of 
proportionality. 

11.  If  the  product  of  two  numbers  is  equal  to  the  product 
of  two  other  numbers,  either  pair  may  be  made  the  means  and 
the  other  pair  the  extremes  in  a  proportion. 

12.  A  proportion  is  obtained  from  a  given  proportion  by 
alternation,  if  the  means  are  intercJianged,  or  if  the  extremes 
are  interchanged. 

13.  One  of  two  numbers  varies  directly  as  the  other  if 
their  ratio  is  constant. 

14.  One  number  varies  inversely  as  another  if  the  ratio 
of  the  first  to  the  reciprocal  of  the  second  is  constant. 


CHAPTER  VII 

PROBLEMS  ON  PARALLEL  LINES.     GEOMETRIC 
CONSTRUCTIONS 

Parallel  Lines 

104.  The  equation  may  be  used  to  solve  exercises  on  the 
angles  made  by  lines  intersecting  parallel  lines. 

1.  Point  out  the  parallel  edges  of  a  rectangular  box  or 
table-top.  Give  other  examples  of  parallel  lines,  as  telegraph 
wires,  latitude  lines,  plumb-lines,  etc. 

2.  Draw  two  parallel  straight  lines,  using  the  parallel  edges 
of  a  ruler;  draw  a  straight  line  inter- 
secting them  (Fig.  152).  Measure  and 
compare  angles  a  and  h,  c  and  d,  e  and 
/,  g  and  h.  These  angle-pairs  are  called 
pairs  of  corresponding  angles. 

3.  Draw  two  figures  like  Fig.  152, 
with    the    crossing-lines    in    different 
directions,    and    compare    the    corre-  Fig.  152 
sponding  angles. 

4.  Draw  two  straight  lines  that  are  not  parallel,  and  com- 

pare the  corresponding  angles  that 
they  make  with  a  crossing-line 
(transversal). 

105.  Parallel  straight  lines  are 
lines  that  have  the  same  direction 
or  opposite  directions. 

I.  :x;  and  y  (Fig.  153)  are  equal. 
Show  by  rotating  a  pencil,  as  in- 
dicated, that  A  B  and  C  D  have 
the  same  direction. 

^37 


138 


First- Year  Mathematics 


2.  Draw  two  intersecting  lines,  AB  and  RS  (Fig.  154). 
Show  how  to  draw  with  the  aid  of  a  protractor  a  line  parallel 
to  A  B  through  M.  X 


A 


J) 


3 


Fig.  154 


Fig.  155 


3,  Draw  a  triangle  on  paper.  Cut  out  the  triangle  and 
place  one  side  of  it  along  a  line  X  Y  (Fig.  155),  first  in  position 
(i)  drawing  line  A  B,  and  then  in  position  (2)  drawing  C  D. 
Show  that  A  B  and  C  D  are  parallel.    What  angles  are  equal  ? 

4.  Fold  a  piece  of  paper  so 
as  to  form  a  right  angle  (r,, 
Fig.  156).  Using  the  right  angle 
as  in  the  figure,  draw  A  B  and 
C  D  and  show  that  they  are 
parallel. 

106.  In  the  preceding  prob- 
lems, the  following  properties  of 
parallel  lines  were  studied: 

(a)  If  two  straight  lines  are 
parallel,  the  corresponding  angles  made  with  a  crossing-line 
(transversal)  are  equal. 

(6)  Two  straight  lines  that  make  equal  corresponding 
angles  with  a  transversal  have  the  same  direction,  and  are 
parallel. 

From  (b)  it  follows  that 

(c)  Two  lines  that  are  perpendicular  to  the  same  line  are 
parallel. 


Problems  on  Parallel  Lines 


139 


1.  In  Fig.  157  read  several  pairs  of  supplementary  adjacent 
angles  (see  p.  42);  of  vertical  angles  (see  p.  47). 

2.  Draw  two  parallel  straight  lines  and  a  transversal. 
Letter  the  angles  as  in  Fig.  157.  Give  reasons  for  the  fol- 
lowing: 

a=e  (i) 

a-\-d  =  \?>o  (2) 

therefore,   e-\-d  =  i^o.  (3) 

Give  reasons: 

3.  If  a=e,   then  c=e  (Fig.  157). 

4.  If  c=e,    then  c=g. 

5.  If  c=g,    then  a  =^. 
6.  Show  that  h  is  the  supple- 
ment of  a;  that /is  the  supplement  of  e. 
a=e. 


Fig.  157 
Show  that  &=/,  if 


7.  Give  reasons  for  the  following: 
b+c  =  iSo 


therefore, 

Give  reasons: 


(i) 
(2) 
(3) 


Fig.  158 


8.  If  b=f,  then  d=f, 

9.  li  d=f,  then  d=h. 
10.  If  d=h,  then  b=h. 

11.  Make  a  list  of  aH  equal  angles  of 
Fig.  157,  and  of  the  pairs  of  non-adjacent 
angles  that  were  proved  to  be  supple- 
mentary. 

12.  Make  a  list  like  11  for  Fig.  158, 
A  B  and  C  D  being  parallel. 

13.  One  of  the  16  angles  made  by  the 
rails  of  two  intersecting  railroads  is  35°. 
Find  each  of  the  other  15. 


I40 


First-Year  Mathematics 


c  and/ 
d  and/ 
c  and  e 


107.  When  two  lines  (Fig.  159)  are  cut  by  a  transversal, 

ia  and  e  \  are  called 
b  and/  (  corre- 
d  and  h  (   spending 
c  and  ^  /  angles, 
angles  c,  d,  e,  f  are  called  interior  angles, 
angles  a,  b,  g,  h  are  called  exterior  angles, 
d  and  e  )  are  called  interior  angles  on  the 
I      same  side  of  the  transversal, 
on  opposite  sides  of  the  transversal, 
are  called  alternate  interior  angles, 
on  opposite  sides  of  the  transversal, 
are     called     alternate     exterior 
angles. 

1.  State  the  results  of  problem  11,  p.  139,  naming  the 
angle-pairs  as  just  defined. 

2.  Draw  a  pair  of  parallels  crossing  another  pair.     Letter 
the  angles  as  in  Fig.  160.     Prove  a^t,  t  =  m;  therefore  a=w. 

H  Z 


Fig.  159 
the  angles  of 
the  angle-pairs 
the  angles  of 
the  angle-pairs 

the  angles  of     )  b  and  h 
the  angle-pairs  )  a  and  g  { 


3.  Prove  that  a -^5-  =  180°.  ■ 

w-f-g=i8o°.     Why?         m=a.     Why?         0  +  9  =  180°.     Why? 

4.  Prove  that  r  is  the  supplement  of  d,  and  that  e  is  the 
supplement  of  </;  therefore,  n^e. 


Problems  on  Parallel  Lines  141 

5.  Prove  the  following:     Fig.  160. 

(i)         d=q  (5)  g=t 

(2)  w=f  (6)     c+q  =  iSo 

(3)  w+g  =  i8o  (7)     a+/  =  i8o 

(4)  b=l  (8)     s+e  =  i8o. 

6.  From    the   top  of   a  cliff  (Fig. 
161),  200  ft.  high,  the  angle  of  depres- 
sion of  a  buoy   is  60°.     Prove   that   — "-"-  -'"-  — ^^ 
angle    B   is  60°.     Find,    by    a    scale  '' 
drawing,    the    distance    of    the    buoy                          >/ 
from  the  bottom  of  the  cliff.                                           i/ 

1 08.  A  quadrilateral  whose  oppo-  ^\ 


site  sides  are    parallel   is  a   parallelo-  b 

gram. 

1.  In  the  parallelogram,  Fig.  162,  prove  that  consecutive 
angles,  as  x  and  y,  y  and  z,  are  supplementary. 

2.  Prove  that   the  opposite  angles,  x  and   z,  y  and  w,  of 
the  parallelogram,  Fig.  162,  are  equal. 

3.  Prove  that  the  sum  of  the  interior  angles  of  a  parallelo- 
/  gram  is  four  right  angles. 


Fig.  162 


4.  With  ruler  and  protractor,  draw  a  parallelogram  having 
adjacent  sides  3  in.  and  5  in.  (Fig.  163),  and  included  angle 
60°.  How  many  degrees  are  there  in  a  ?  In  e  ?  In  &  ?  In 
c?    Inrf? 


142  First- Year  Mathematics 

5.  Draw  parallelograms  having  the  following  parts.     Find 
the  number  of  degrees  in  each  of  the  remaining  interior  angles: 
Adjacent  Sides  Included  Angle 

(i)  3  in.  and  5  in.  and  120° 
(2)  3  in.  and  4  in.  and    90° 
(5)  3  in.  and  3  in.  and     40° 
(4)  3  in.  and  3  in.  and    90° 
109.  The  following  theorems  have  been  studied: 
Theorem  I.     Two  consecutive  angles  of  a  parallelogram  are 
supplementary,  and  the  opposite  angles  are  equal. 

Theorem  II.  The  sum  of  the  interior  angles  of  a  parallelo- 
gram is  four  right  angles. 

1 10.^  A  quadrilateral  with  one  pair  cf  parallel  sides  is  a 
^ — j^  trapezoid. 

^^ ^^^  I.  In  the  trapezoid,  Fig.  164,  prove  that 

Fig.  164  X  and  y  are  supplementary. 

2.  Prove  that  the  sum  of  the  interior  angles  of  a  trapezoid 
equals  four  right  angles. 

3.  Draw  a  trapezoid  A  B  C  D  from  the  data  of  Fig.  165. 
Prolong  B  C  and  A  D  until  they  meet  at  O.  Show  that  the 
angles  of  triangle  O  C  D  are  equal  to  the  corresponding  angles 
of  triangle  O  B  A,  and  that  the  two  triangles  are  similar. 

4.  In  Fig.  165,  show  that  O  C:0  B=0  D:0  A,  and  that 
CD:BA=OC:OB. 


Problems  on  Parallel  Lines 


143 


J .r^r. 


5.  Draw  a  triangle  ABC  (Fig.  166)  having  an  angle  of 
60°.  Draw  lines  B  X  and  C  X,  making  angles  as  indicated. 
Prove  that  A  B  X  C  is  a  parallelogram. 

6.  Draw  a  parallelogram,  starting  with  a  triangle  having 
an  angle  of  135°;  an  angle  of  90°.  / 

7.  Using    Fig.    167,    in    which  / 
D  E  is  parallel  to  A  C,  prove  that 
the  sum  of  the  interior  angles  of  a 
triangle  is  two  right  angles. 

8.  What  is  the  sum  of  the  ex- 
terior angles  (formed  by  prolonging 
a  side)  of  a  triangle  taking  one  at  each  vertex  ? 

9.  From  Fig.  168,  prove  that  an 

^r\ ^     exterior  angle  of  a  triangle  (formed 

Fig.  168  by  prolonging   a   side)  equals  the 

sum  of  the  interior  angles  not  adjacent  to  it. 


Fig.  167 


Algebraic  Exercises  on  Geometric  Figures 

I.  Two  parallels  and  a  transversal  make  angles  that  may 
be  designated  as  shown  in  Fig.  169.     Find  the  values  of  x,  and 

of  all  the  unknown  angles. 


Fig.  169 


Fig.  170 


2.  Two  parallels  and  a  transversal  form  angles  that  may 

V be  designated  as  shown  in  Fig.  170.     Find 

^^'  :x;  and  all  the  8  angles. 

\\  3.  With  parallels,  and  angles  as  shown 

Fig.  171  in  Fig.  171,  find  x  and  all  the  8  angles. 


144 


First-Year  Mathematics 


4.  Two  parallels  are  cut  by  a  transversal  making  a  pair  of 
alternate  interior  angles  of  5^+3°  and  5g°—3y.  Find  y  and 
all  the  8  angles. 

5.  With  two  parallels  and  a  transversal,  a  pair  of  corre- 
sponding angles  are  x-\-2y  degrees  and  2{x—y)  degrees,  the 
angle  adjacent  to  the  latter  being  120°.  Find  x,  y,  and  the 
unknown  angles. 

6.  With  two  parallels  and  a  transversal,  a  pair  of  alternate 
exterior  angles  are  5y  —  2X,  and  gx+y.  The  angle  adjacent 
to  the  latter  is  86°.     Find  x,  y,  and  the  unknown  angles. 

7.  With  two  parallels,  the  interior  angles  on  the  same  side 
of  a  transversal  are  6x+y  degrees  and  1451;—^  degrees  and  their 
diflference  is  14°.     Find  x,  y,  and  all  the  8  angles. 

8.  With  two  parallels,  the  interior  angles  on  the  same  side 
of  the  transversal  are  <^x  —  ya.nd  5(2^+^).  ^{2y-\-x)  and  125° 
are  alternate  interior  angles.  Find  x,  y,  and  the  unknown 
angles. 

9.  With  parallels,  transversal,  and  angles  as  shown  in 
Fig.  172,  find  X,  y,  and  all  the  8  angles. 


Fig.  172 


Fig.  173 


10.  With  parallels,  transversal,  and  angles  as  shown  in 
Fig.  173,  find  X,  y,  and  all  the  8  angles. 

11.  With  two  parallels  and  a  transversal,  the  alternate  ex- 
terior angles  are  7(:x:  +  i)  degrees  and  181  — 2:x;  degrees.  Find 
X  and  all  the  8  angles. 


Problems  on  Parallel  Lines  145 

Problems  on  Construction 
III.  The  following  geometrical  figures  are  to  be  constructed 
by  means  of  compasses  and  unmarked  straight  edge  only. 

1.  To  construct  a  line  perpendicular  to  a  given  line,  AB,  at 
a  given  point,  C,  on  the  line  AB. 

Construction:    With    C    (Fig.  I 

174)    as    center    and    a    convenient  I 

radius,  draw  arcs  of  a  circle  meeting  >♦* 

A  B  in  two  points  as  D  and  E.  I 

With  D  and  E  as  centers  and  a  ,         I         | 

convenient  radius,  draw  arcs  of  two  •" 
circles  meeting  as  at  F  and  G. 

Connect  by  a  straight  line  one  of  ^^ 

the  points  as  F  with  C.  ^^^-  ^74 

The  line  F  C  is  the  required  perpendicular  to  A  B  at  point  C. 

How  long  must  the  radius  D  F  be  ? 

2.  Construct  a  right  triangle  having  given  the  two  sides, 
3  in.  and  4  in.,  which  meet  at  the  vertex  of  the  right  angle. 

A  rectangle  is  a  parallelogram  one  of  whose  angles  is  a  right 
angle. 

3.  To  construct  a  rectangle,   having  given  the  length  / 
and  the  width  w. 

• 1 Construction:   On  any  convenient 

■ ^ —  line,  A  B  (Fig.  175),  lay  off  C  D  equal  in 

length  to  /. 

At  C  and  D  construct  the  perpendic- 
ulars C  E  and  D  F. 

On  C  E  and  D  F  lay  off  the  lengths 
C  G  and  D  H  each  equal  to  w. 
Draw  G  H. 

The   quadrilateral  C  G  H  D  is   the 
Fig.  175  required  rectangle. 

4.  Construct  the  complement  of  a  given  acute  angle  ABC. 

5.  Construct  a  square  having  given  one  side  s. 

6.  Construct  two  lines  perpendicular  to  a  given  line  and 
prove  the  two  lines  parallel. 


s 

F 

-•c 

~' 

• 

l« 

W 

A. 

c 

»                         1 

146  First- Year  Mathematics 

7.  Draw  a  right  triangle.     Construct  a  rectangle  having 
for  two  of  its  sides  two  sides  of  the  triangle. 

8.  From  a  point,  C,  outside  of  a  line,  AB,  fo  construct  a 
line  perpendicular  /o  AB. 

Construct  CF  (Fig.  176)  perpendicular 
I  to  A  B,  following  the  directions  given  in  the 

f^  construction  for  problem  i. 


:k 


A ^ 


Fig.  176  Fig.  177 

9.  Construct  a  perpendicular  from  C  to  A  B  when  C  and 
A  B  are  as  in  Fig.  177. 

10.  Draw  a  triangle  of  which  all  angles  are  acute.     From 

the  vertex  of  each  angle  construct  a  line  perpendicular  to  the 

opposite  side. 

It  is  proved  in  geometry  that  the  three  altitudes  of  a  triangle  meet 
in  a  point.  This  gives  a  test  of  the  accuracy  of  the  constructions  in 
problems  10,  11,  and  12.  Crease  the  three  altitudes  of  a  paper  tri- 
angle and  find  whether  they  all  pass  through  the  same  point. 

11.  Draw  a  triangle  having  an  obtuse  angle  and  construct 
perpendiculars  as  in  problem  10. 

12.  Draw  a  right  triangle  and  construct  perpendiculars  as 
in  problem  10. 

13.  Draw  a  parallelogram,  as  A  B  C  D  (Fig.  178).  Con- 
struct the  altitudes  from  B  and  C  to  the  base  A  D ;  from  D 
to  the  base  A  B. 


Fig.  179 

14.  Angle  F  (Fig.  179)  is  a  right  angle  and  h  is  the  altitude 
from  F  to  D  E.     Find  the  values  of  x,  y,  and  z. 


Problems  on  Parallel  Lines  147 

15.  To  bisect  a  given  line,  AB. 

Construction:    Using  the  end  points  A  and  B  as  the  points  D 
and  E  were  used  in  problem  i,  draw  a  line, 
as  F  G,  perpendicular  to  A  B.  \^ 

The    point    of    intersection    H    is   the  I 

required  midpoint  of  A  B. 


The  line  F  G  is  called  the  perpendicular      ■**  i^ 


bisector  of  A  B. 

16.  Draw  a    triangle.     Bisect  the  )*^ 
sides  and  join  the  midpoint  of  each                Fig.  180 
side  to  the   vertex  of   the   opposite   angle. 

A  straight  line  drawn  from  the  midpoint  of  a  side  of  a 
triangle  to  the  vertex  of  the  opposite  angle  is  a- median. 

Show,  by  creasing  the  3  medians  of  a  paper  triangle,  how 
the  accuracy  of  the  construction  of  the  three  medians  of  a 
triangle  may  be  tested. 

17.  Draw  a  right  triangle.  Construct  the  medians  and 
show,  by  drawing  a  circle  on  the  hypotenuse  ao  a  diameter, 
that  the  median  to  the  hjrpotenuse  equals  one-half  of  the 
hypotenuse. 

18.  Draw  a  right  triangle,  a  triangle  having  an  obtuse 
angle,  and  a  triangle  of  which  all  angles  are  acute.  In  each 
of  the  triangles  construct  the  perpendicular  bisectors  of  the 
three  sides.  What  seems  to  be  the  test  for  the  accuracy  of 
the  construction?  Crease  the  perpendicular  bisectors  of  the 
three  sides  of  a  paper  triangle  and  see  whether  the  test  is  met. 

19.  To  bisect  a  given  angle  ABC. 
Construction:   With  B  -as  center  and 
any  radius,  draw  arcs  intersecting  B  A  and 
B  C  in  two  points  as  D  and  E. 

With  D  and  E  as  centers  and  a  con- 
venient radius,  draw  arcs  meeting  as  at  F 
Fig.  181  and  G. 

Connect  by  a  straight  line  one  of  these  points,  as  F,  with  B. 
The  line  B  F  is  the  required  bisector  of  angle  ABC. 


148 


First-Year  Mathematics 


20.  Draw  a  triangle.     Construct  the  bisectors  of  the  three 
angles  of  the  triangle.    What  seems  to  be  a  test  for  the  accu- 
racy of  the  construction  ?    Find  a  test  for  the  accuracy  of  the 
L  g  construction  by  creasing  the  bisect- 

ors  of  the  angles  of  a  paper  tri- 
angle. 

21.  Construct  the  bisectors  (LO 
"v4  and   M  O)   of    two    supplementary 
^^^"  adjacent  angles  (Fig.  182).     Prove 

that  the  bisectors  are  perpendicular  to  each  other. 

22.  Construct  the  bisectors  of  a  pair  of  corresponding 
angles  made  by  a  transversal  cutting  two  parallels,  Fig,  183. 
Prove  that  the  bisectors  are  parallel. 


Fig.  184 


Fig.  183 


23.  Construct  the  bisectors  of  a  pair  of  interior  angles  on 
the  same  side  of  a  transversal  cutting  two  parallels  (Fig.  184). 
Prove  that  the  bisectors  are  perpendicular  to  each  other. 

24.  At  a  given  point,  A,  on  a  line,  B  C,  to  construct  an 
angle  equal  to  a  given  angle,  D  E  F. 

Construction:  With  E  as  center  (Fig.  185)  and  any  radius,  draw 
an  arc  meeting  E  F  and  E  D  in  the  points,  G  and  H. 

With  A  as  center  and  the  same  radius,  draw  the  arc  M  N. 


Problems  on  Parallel  Lines 


149 


With  M  as  center  and  radius  equal  to  the  distance  from  H  to  G, 
draw  an  arc  meeting  M  N  at  O. 

Draw  A  O. 

Then  O  A  M  is  the  required 
angle. 

25.  Draw  a  line  parallel  to 
a  given  line  passing  through  a 
point  outside  of  the  given  line. 

Construction  is  as  in  problem  2, 
p.  138.  Instead  of  using  the  pro- 
tractor to  construct  the  angle,  ruler 
and  compasses  are  to  be  used. 

26.  Draw  a   triangle.     Con- 
struct   an  angle    equal   to    the 
sum  of  the  angles  of  the  triangle, 
accuracy  of  the  construction  ? 

27.  Draw  a  triangle.     Construct  a  parallelogram   having 
for  two  of  its  sides  two  sides  of  the  triangle. 

28.  To  construct  triangles  having  the  same  base  and  equal 
areas. 

Construction:   Construct  a  line  C  D  (Fig.  186)  parallel  to  A  B. 


Fig.  185 
How  can  you  test  the 


^ 


ms^ 


z^ 


Fig.  186 


Connect,  by  straight  lines,  the  points  A  and  B  with  points  on  C  D 
as  E,  F,  G,  etc. 

Show  that  the  triangles  thus  formed  have  equal  altitudes  and  equal 
areas. 

29.  Draw  a  triangle.  Construct  another  triangle  having 
two  sides  and  the  included  angle  equal  to  two  sides  and  the 
included  angle  of  the  first.  Compare  the  triangles  as  to  size 
by  cutting  out  one  of  them  and  fitting  it  on  the  other. 


ISO 


First-Year  Mathematics 


30.  Draw  a  triangle.  Construct  another  triangle  having 
one  side  and  the  two  angles  adjacent  to  that  side  equal  to  a 
side  and  its  adjacent  angles  of  the  first.  Compare  the  triangles 
as  to  size. 


Algebraic  Exercises  on  Geometric  Figures 

I.  The  angles  made  by  two  pairs  of  parallels  intersecting 
as  in  Fig.  187  are  designated  as  shown.  Find  x,  y,  and  all 
4  angles  about  the  crossing-point,  K. 


Fig.  187 

2.  With  two  pairs  of  intersecting  parallels  and  angles  as 
shown  in  Fig.  188,  find  x,  z,  and  all  4  angles  around  any 
crossing-point. 


Fig.  I 

3.  With  the  sides  and  angles  as  shown  in  Fig.  189,  find  y, 
z,  a,  and  b,  and  all  the  4  angles  and  4  sides  of  the  parallelo- 
gram. Assume  that  the  opposite  sides  of  a  parallelogram  are 
equal. 

5a  .3 


I 


"SF 


Fig.  190 
Fig.  189 
4.  In  the   trapezoid  of  Fig.   190,  find  x,  y,   and  all 


angles. 


Problems  on  Parallel  Lines 


151 


5.  In  the  trapezoid,  Fig.  191,  find  x,  y,  and  all  4  interior 
angles.  f^ 

6.  The  non-parallel  sides  of  a       / 
trapezoid   are   extended  until  they      /^y      ■  y^ 
meet,    Fig.    192.     The   lengths    of                   Fig.  191 

lines    being    designated   as  shown,   find  x 
and  y. 

7.  The  non-parallel  sides  of  a  trapezoid 
being  prolonged  to  intersect  and  the  lengths 
of  lines  being  designated  as  shown  in  Fig. 
iQ^,  find  X  and  y  and  f,^-^^» 

€r ^ 


Fig.  192 
the  lengths  of  all  lines. 

8.  The  non-parallel  sides  of  a  trape- 


FiG.  194 


Fig.  193 

zoid  are  extended  to  meet,  Fig.  194.  Find 
the  lengths  of  lines,  h,  c,  and  the  area  of 
the  trapezoid  in  terms  of  a. 

9.  Find  all  angles  of  Fig.  194,  lengths  of 
lines  being  as  shown  and  the  angle  at  the 
top,  53°  8'. 


Summary 

1.  Parallel  straight   lines  are  lines  that   have  the  same 
direction  or  opposite  directions. 

2.  If  two  lines  are  cut  by  a  transversal,  making  a  pair  of 
corresponding  angles  equal,  the  lines  are  parallel. 

3.  Two  lines  that  are  perpendicular  to  the  same  straight 
line  are  parallel. 

4.  If  two  parallel  lines  are  cut  by  a  transversal, 

(i)  the  corresponding  angles  are  equal; 

(2)  the  alternate-interior  angles  are  equal; 

(3)  the  sum  of  the  interior  angles  on  the  same  side  of  the 
transversal  is  two  right  angles. 


152 


First-Year  Mathematics 


5.  A  parallelogram  is  a  quadrilateral  whose  opposite  sides 
are  parallel. 

6.  Two  consecutive  angles  of  a  parallelogram  are  supple- 
mentary, and  the  opposite  angles  are  equal. 

7.  The  sum  of  the  interior  angles  of  a  parallelogram  is 
four  right  angles. 

8.  An  exterior  angle  of  a  triangle  (formed  by  prolonging  a 
side)  is  equal  to  the  sum  of  the  two  interior  angles  not  adjacent 
to  it. 

9.  A  trapezoid  is  a  quadrilateral  with  one  pair  of  parallel 
sides. 

10.  The  sum  of  the  interior  angles  of  a  trapezoid  is  four 
right  angles. 

11.  To  construct  figures  by  the  aid  of  compasses  and 
unmarked  straight  edge  only,  the  following  fundamental  con- 
structions are  used: 

(i)  To  construct  a  perpendictdar  to  a  line  at  a  point  on  the 
line; 

(2)  To  construct  a  perpendicular  to  a  line  from  a  point 
without; 

(3)  To  bisect  a  line; 

(4)  To  bisect  an  angle; 

(5)  To  construct  an  angle  equal  to  a  given  angle. 

12.  A  median  of  a  triangle  is  a  straight  line  drawn  from 
the  mid-point  of  a  side  to  the  vertex  of  the  opposite  angle. 


CHAPTER  VIII 

THE  FUNDAMENTAL  OPERATIONS  APPLIED  TO  INTEGRAL 
ALGEBRAIC  EXPRESSIONS 

Addition  of  Monomials 

112.  The  tickets  for  a  football  game  are  sold  at  25^  by 

John,  Henry,   Kenneth,  William,  and  James.     They  report 

sales  as  follows:    John  sold  56  tickets,  Henry  75,  Kenneth  27, 

William  83,  James,  69.     At  the  gate   123  tickets  are  sold. 

Find  the  total  receipts. 

Solution  I: 

John,  56X25^=$i4.oo 

Henry,  75X25^=   18.75 
Kenneth,  27X25^=     6.75 

William,  83X25!*=   20.75 

James,  69X25/=   17.25 

Gate,  123X25/=  30.75 

Total  receipts  $108 .  25 

Solution  II: 

John,  56  tickets,  56X25/ 

Henry,  75  tickets,  75  X  25/ 

Kenneth,  27  tickets,  27X25J* 

William,  83  tickets,  83X25/ 

James,  69  tickets,  69X25/ 

Gate,  123  tickets,  123X25/ 

433  tickets,  433X25/ =  $108. 25. 

In  Solution  II  the  addition  is  simplified,  because  the  dif- 
ferent terms  to  be  added  have  a  common  factor  and  they  were 
therefore  added  by  adding  the  coefficients  of  this  factor  in  the 
different  terms. 

I.  The  tickets  being  sold  at  a;^,  John  sells  60  tickets, 
Henry  78,  Kenneth  45,  William  36,  and  James  84.  At  the 
gate  137  tickets  are  sold.     Find  the  total  receipts. 

J.      K       ^      g        I       c! 
Total  receipts:     6ox->r^?>x-\■^^x-\■^,6x  +  ?>4x-'^li'JX  =  ^/^ox. 

153 


154  First-Year  Mathematics 

Again  the  different  terms  to  be  added  had  a  common 
factor  and  the  polynomial  resulting  from  the  addition  could 
be  simplified  by  adding  the  coefficients  of  the  common  factor  in 
the  different  terms. 

Is  Solution  I  of  p.  153  possible  in  this  case?  Give  a 
reason  for  your  answer. 

2.  I  wish  to  change  the  following  German  currency  to 
American  money:  10  five-mark  pieces,  17  one-mark  pieces, 
28  twenty-five-mark  bills,  and  123  fifty-mark  bills.  The  value 
of  a  mark  is  23.8^.  How  much  American  money  should  I 
receive  in  exchange  ? 

3.  The  length  of  the  school  hall  is  5  feet.  I  go  through 
the  hall  6  times  on  Monday,  8  times  on  Tuesday,  4  times  on 
Wednesday,  6  times  on  Thursday,  and  10  times  on  Friday. 
How  many  feet  do  I  travel  along  the  hall  during  the 
week? 

4.  The  running  track  in  the  playground  is  y  yards.  While 
in  training,  I  take  it  6  times  on  Monday,  8  times  on  Tuesday, 
10  times  on  Wednesday,  12  times  on  Thursday,  and  14  times 
on  Friday.  How  many  yards  do  I  run  during  the 
week? 

113.  Terms  which  have  a  common  factor  are  called  similar 
with  respect  to  that  factor. 

I.  Point  out  with  respect  to  what  factor  the  following  terms 
are  similar.  State  in  each  case  the  coefficient  of  the  common 
factor: 

(i)  4X,  -yx,  2o:c,  -S5X 

(2)  ax,  —2^x,  —bx,  463c 

(3)  ax,  —bx,  —ex,  dx 

(4)  2xa,  yea,  -jxa,  —sxa 

(5)  -3Pi',  6/?%  -Srq^,  i2,sq' 

(6)  4axz,  —ycxz,  -sdxz,  gexz 

(7)  abm,  —pmq,  —xmy,  —mdc 


Fundamental  Operations  and  Algebraic  Expressions    155 

(8)  sa'b,  -i^ab',  +4a^6%  -i.5a3& 

(9)  ax\  s^xy,  -y.sxt,  -ijbx' 

(10)   —4at3,  —abt^,  -^zaxP,  +'j.2ayt\ 

2,  State  a  rule  for  simplifying  polynomials  consisting  of 
similar  terms. 

Exercise  XV 

Express  the  sums  of  the  following  numbers  in  the  form  of 
a  polynomial  and  simplify: 

1.  +i5a,  -7a,  +i8a 

2.  —iSx',  —X2X^,  +15X',  —sx' 

3.  +2iab,  +3ja6,  -4ia6,  -5§a6 

4.  +2'jabc,  —  35a6c,  -\-ioabc,  —2abc 

5.  s(a+b),  -4(a+b),  i2{a+b) 

6.  -^x'+y'),  -24{x'+y'),  i^ix^+y") 

7.  -3Kpr-q'),  +5Upr-q'),  -4Mpr-q') 

8.  iS{mp-3sy,   -i5{mp-yy,  -37{mp-3sy, 
iA{mp-3s)' 

9.  a{x-\-y-{-z),  -b{x+y+z),  -c{x-\-yi-z),  d{x^y+z) 
10.  isax',  —ybx',  Sdx',  —^cx'. 

114.  At  a  money  changer's  the  following  amounts  of 
money  are  offered  for  exchange:  527  marks,  349  francs,  57 
pounds.  The  exchange  value  of  a  mark,  a  franc,  and  a 
pound  are  23.8^,  19.3^,  and  $4.87  respectively.  What  is  the 
exchange  value  of  the  foreign  currency. 

527  marks     at  23.8)''  =  527X23.8^  =  $i25.426 

349  francs     at  19. 3^'' =  349X19. 3^=     67.357 

57  pounds   at  $4.87=  57X$4-87=  277.59 

$470-373 

Could  the  method  of  Solution  II  on  p.  153  have  been 

applied  here  ? 

I.  A  beam  is  loaded  as  indicated  below.     Find  the  total 

turning-tendency  in  each  case: 


156 


First-Year  Mathematics 


/. 

rf. 

h 

d. 

h 

ds 

/4 

^4 

I 

+3 

-17 

-  5 

-14 

-  8 

-5 

+  12 

+  19 

II 

+5 

—a 

+  12 

-b 

+  14 

+6 

,..  6 

+a 

III 

-8 

—X 

+  IO 

-y 

-IS 

—2 

+  9 

+  7f 

2.  The  main  stairway  in  a  school  consists  of  four  flights 
of  stairs,  having  a,  b,  c,  and  d  steps,  respectively.  If  a  pupil 
goes  up  and  down  8  times  in  one  day,  how  many  steps  does 
he  take  on  that  stairway  daily  ? 

115.  Terms  which  have  no  common  factor  are  called 
dissimilar  terms. 

Can  a  polynomial,  whose  terms  are  not  all  similar,  be 
simplified  into  one  term  ? 

Exercise  XVI 

Add  the  following  numbers  and  simplify  the  resulting  poly- 
nomials as  far  as  possible. 

1.  55%  —i2X''y,  +yx''y,  —ss't 

2.  — 27fa6,  +iSyd,  isiab,  i^^cd 

3.  sax,  -3^,  X 

4.  9a*6%  -3(^^y^,  4a'6^  -4c3;y3,  -^a'b'' 

5.  3w/»%  -8w/>^  +5a%  -sa^'x,  -^mp\  2a^x 

6.  -4st,  -t,  3/,  +55/ 

7.  ar^'p,  -r'p,  -br'p,  pr',  -cpr"" 

8.  27(a+&)%  +4C,  +150,  -I2C,  -i5(a+6)%  -8a 

9.  a(a-6),  b{a-b),  -c(a-b) 
lo.  a^'Ca+fc),  -2a6(a+6),  6^(a+6) 

n.  -6Ka+6),  K<i'+6"),   -i(a^+6^),   +8Ka  +  fc), 

+4Ka+5),  6J(a^+6^) 
12.  r3(r+/)),  -3r-(r+/,),  sr{r+p),  r-^-p. 


Fundamental  Operations  and  Algebraic  Expressions    157 

Addition  of  Polynomials 
116.  Consider  problem  2,  p.  156.     If  a  pupil  goes  up  and 
down  the  stairs  5  times  on  Monday,  6  times  on  Tuesday,  4  times 
on  Wednesday,  5  times  on  Thursday,  and  4  times  on  Friday, 
how  many  steps  does  he  take  on  the  stairway  every  week  ? 

On  Monday:        loa+iob  +  ioc  +  iod 

On  Tuesday:        I2a+i2b  +  i2c  +  i2d 

On  Wednesday:     8a  +  8b  +  8c  +  8d 

On  Thursday:      loa  +  iob  +  ioc+iod 

On  Friday:  8a+  8b+  8c+  8d 

4.8a  +  48b  +  48c +  48d 

We  add  these  5  polynomials  by  adding  the  similar  terms  in  them 

I.  Add:  +2'jx'  —  i$xy  +  iSy' 

—  i2X^+2oxy—^y' 
Adding  the  similar  terms  in  the  polynomials: 
+ 15^*  +  T^Sxy  +  i$y^. 
Frequently  the  work  is  arranged  along  one  line  as  follows. 
{  +  2jx^  —  i^xy  +  i8y^)  +  {  —  i2x^  +  ^oxy  —  ^y). 
Adding  similar  terms:   i^x'  +  i$xy  +  i^y^. 

Exercise  XVII 
Add  the  following  polynomials  and  simplify: 

1.  ^a-\-2,b-\-gc,  2a  — 10b  — 2c 

2.  6a  +  isc— 17&— 8(/,  —'ja  +  2id-]-i$b  —  i2c 

3.  7a*  +  236^  — i5c%  —2162—6^^4- 12c* 
*4.  2gxy  —  'jy'  +  24X',  —i4xy—^6x^  +  7,6y' 

5.  — 3a  — 76  +  14C,  — iia  +  206— 34c 

*6.  14^  — n/  +  i2w,  —3^  +  12/— 6w,  — 12^+/  — 2W 

7.  -iSa''b''-\-i2a^-Sa3b,  4ja3&-a='6^+3fa4 

8-  ^x^'  —  ^xy—ly',  —x'  —  ^xy  +  2y'',^x''—xy  —  ^y'' 

9.  s(a+6)-7(a^+6»)+8(a3+63),  _4(o3+fe3)+5(a=>+6a) 

*io.  |(a  +  6  +  c)-|(a-&+c)+|(a+6-c)  +  iK-a+&+c), 
-f(fl+&+c)+f(a-&+c)-f(a+6-c)-f(-a+&+c) 

♦  The  exercises  marked  with  a  *  should  be  arranged  and  solved  in 
both  ways. 


1^8  First-Year  Mathematics 

II.  P^+3P'+^P-^'  -p'-2p  +  i,  p'-i,  spi  +  2p  +  2       . 
*i2.  x'+2xy+y',  x'-2xy-\-y',  x'—4xy,  4xy+y' 

13.  —  23a^6  +4ia^c4-56c»6  — 156%  — 6a^6  +  26a*c  +  59c^& 
—266%  2sa^c+igb''c—iSc'b 

14.  5a3{a+b)  -6a'b{a' -\-b')  -\-sab'{a3+b3), 

a'b{a'-\-b')+4a3{a+b)-jab'{a3-{-b3) 
*i5.  6(/r+/)+7(/-M)+fv,  5iv-S(lr-{-t)-s(l-u), 

S{lr-\-t)-(l-u)-4tv. 

Subtraction  of  Monomials 

117.  It  has  been  shown  (see  pp.  75,  76)  how  subtraction 
of  algebraic  numbers  may  always  be  changed  into  addition. 

Example:  +5^  may  be  replaced  by  +  5* 
—  jx  +7X 
S  A 

Subtract  the  lower  monomials  from  the  upper: 
I.  +155;^  2.  -7ia  3.  -15  ab       4.  -  S^{p-\-q) 

-17X'  +3ia  -i8§a6  +i4Kp+q) 

5.  +sm'px  7.  -7Ux-y) 
-Atn'px  -S\{x-y) 

6.  +i4:x;(i+5a^>')  8.   +iiw^(a  — 26') 
+  %x{i-\-^a^y)  +2gm^{a  —  2b^) 


1 18.  Instead  of  writing  the  subtrahend  under  the  minuend, 
it  is  often  written  on  the  same  line  with  the  minuend,  con- 
nected with  it  by  a  minus  sign,  — ,  to  indicate  subtraction,  thus: 

(  +  5x)  — (  — 7*)  is  equivalent  to  (  +  5«)  +  (  +  7x)  or  +$x+7x  or 
+  i23t:.     Why? 

Omitting  the  second  step  the  work  may  be  written  as  follows: 
(  +  S^)-(-7^)  =  +5«+7»=+i2x. 

*  The  exercises  marked  with  a  *  should  be  arranged  and  solved  in 
both  ways. 


Fundamental  Operations  and  Algebraic  Expressions    159 

Exercise  XVIII 
Simplify: 

1.  (+506)  — (  +  i2a6) 

2.  (-75a^r/^)-(-54a^r/=') 

3.  (-l8/»3/3)_(+63/3/»3) 

5-  (— siM^t')— (  — 2.4M^z') 

6.  (  +  8.7/>35*54)-(-4^/>3^^54) 

7.  |-5(a+6)f-|+7(<x+&)} 

8.  |+4(f^_/3)[-|_2(/^-/3)| 

9.  \-6{m'+g)\-\-a{m^^g)\ 

10.  |— :»(?;='— 5*) |—j+7(^'''—^^) I 

11.  \a{x-\-y)\-\h{x-\-y)\ 

12.  I  — 2ia(:!c^— a>')f  —  I +4g(ic^— a>')| 

13.  |+i8Ka+&+c)f-H25Ka+&+c)} 

14.  1-3.4/^^'-^')} -I -4-5^^^^-^^)^ 

Subtraction  of  Polynomials 

1 19.  When  the  subtrahend  consists  of  more  than  one  term, 
the  subtraction  may  be  performed  by  subtracting  each  term  of 
the  subtrahend  from  the  minuend. 

For  example,  when  we  wish  to  subtract  7  dollars,  4  quarters, 
and  10  dimes  from  15  dollars,  8  quarters,  and  30  dimes,  we 
subtract  7  dollars  from  15  dollars,  leaving  8  dollars;  4  quar- 
ters from  8  quarters,  leaving  4  quarters;  and  10  dimes  from 
30  dimes,  leaving  20  dimes. 

The  subtraction  of  algebraic  polynomials  is  then  not 
different  from  the  subtraction  of  monomials  and  may  again 
be  reduced  to  addition. 

Example  : 
+  gaf *  —  I  /^y  —  1 2y*   may  be  replaced  by  +  9*^  —  i  \xy  —  i  ay* 
—  7**+  i^y—'i-^y^  +7«*—  5a:y+i5y» 


i6o  First-Year  Mathematics 

1.  Subtract  the  lower  from  the  upper  polynomial: 

(i)  a*4-2a6+6»         (2)      a^h''-\-a*-¥-T,a^h-\-^ah^ 
a'  —  2ab+b'  —a'b'—a^—b*—ia3b—4ab^ 

(3)  x3^T,x''y+sxy'-{-  y3 

2.  Without  rewriting,  subtract  the  upper  polynomial  from 
the  lower  in  the  preceding  problem. 

3.  From  c3_2a^f—(i3—;-2  subtract  —a'c—^d^—r'—c^. 

4.  From    —  ijx* -{- lOy* -\- 4x3y  —  2gxy^  —  ^^x'y'    subtract 
+  i$y^—4ix'y'-^Sx^y—iSx'*—2^xy3. 

5.  Again  the  subtraction  of  polynomials  may  be  arranged 
as  described  in  §118,  p.  158. 

Example:  Instead  of  writing: 

—  4026  +  7062  — 1203  —  2563 

—  18026  +  9062+  503—  463 
S,  we  write: 

(—4026  + 7062  —  I203  — 2563) —  ( —18026 +  9062 +5a3  —  4b3),  which   is 

equivalent  to: 

( —  4026  +  7062  —  1 203  —  2563)  +  ( + 18026  —  9062  —  503  +  4b3) 
or:   —  4026  +  7062  —  1 203  —  2563  + 18026 — 9062  —  503 + 463 
or:  —1703 +  14026— 2062  — 2163.     Why? 

Leaving  out  the  second  step,  the  work  may  be  written: 

(—4026  +  7062  — 1203  —  2563)  — (  — 18026  +  9062  +  503-463)  = 
—  4026  +  7062-1203  —  2563  +  18026-9062  —  503  +  463  = 
—  1703  +  14026  — 2062  — 2163. 

Exercise  XIX 
Simplify: 

1.  {  +  iyx^  —  i4xy—i^y^)  —  {—i6x^-\-i2xy—gy') 

2.  (—sa^x-{-iobxy-\-24b'y—iSaxy)  —  {—6b'y-\-i2axy 

—4a'x—2bxy) 

3.  {-gm''pq-5m3p  —  i4m^q')-{-6m3p-iom'pq 

-iSni'q^) 


Fundamental  Operations  and  Algebraic  Expressions    i6i 

4.  {s^abc-'j^a^b-S^b^c-6^c'a)-i4^a^b-s^c'a 

+  3iabc  +  jiib'c) 

5.  (4$x^y'  —  2'jx'*  +8i_'v4) .- (73x4  +45y'*-\-6sx^y^) 

6.  (+/^x^  —3X'y  +  i2y3  —  jxy')  —  {2y3  —  T,xy'  +4X'y-\-'jx^) 

7.  (i/3  -3^/w'  +4iw3  — 3/^w)  —  (f/w^  +  5i/^w  — 2/3  — 3^3) 

8.  (3  .  4t'354  —  5  .  7i;453  -(-  Q  .  SV^S^)  —  (  —  I  .  "jV^S^  —  3  .  2Z''^5^ 

-AV^s) 

9.  (27:x;3— 6x^j+8y3)  — (— i5x3  4-8:x;;y^-4}'3) 

10.  (4^4-7i/^)-(3i^V-7-6^''  +  5§^V^+6/3) 

+  25A4) 

12.  (  — 2^^3712/ -1-7. ^^2;«2/_ 3. 24^m3/)  —  {2,\kmH—T).tkmH^ 
—  ^.4k3ml-\-6kml^) 

13.  (3a5c3  — 45)'z3)  _-  (_26:x;3  +4^3  —  52^  +  3/;yz3) 

14.  (— 5wy^  — 3mx'M+4m«^)  —  (+3'y^— 6mMz;— 4«Mz; 

120.  Compare  the  signs  of  the  terms  of  the  subtrahend  in 
the  separate  exercises  of  Exercise  XIX,  before  and  after  the 
parentheses  are  removed. 

1.  State  a  rule  as  to  the  effect  of  a  minus  sign  in  front  of 
a  polynomial  in  parentheses. 

2.  State  a  similar  rule  as  to  the  effect  of  a  plus  sign. 

Exercise  XX 

Perform  the  following  operations  and  simplify  results: 

1.  4-\5-(st'-4)\ 

2.  ^-\4k^-^k^-sh\ 


3.  i6e»-|-42e='-3e^  +  2f-(5oe='+3) 

4.  2/-[6/-3g-4/-(2g-4/)] 

5.  4^2_|^._3p+3^a_f3[ 

6.  9x-\sy-{(iy^iz)-{iy-Az)\ 

7.  -3a4+[4a3_(3a3_5a=')-(4a^+3a)] 


i62  Fir  St- Year  Mathematics 


8.  7-5/''-[3-4/>^-4-2/'^  +  i-6/>^-3-4r-4-5/'l 

9.  7jii-\2p^^-ki-p'^-r\-[p^-r-{ki-\-r)\ 

10.  ioss^-iAst-\-^^-\-Ust-2t'-{Ss^-2st)\ 

11.  -hrl-{zr'-2^')\-\Ar'-Srl-2l'\-^{2r-^-2l') 

12.  -|5i*='-(3.4*^-2/i')[-l5-4^/^-3*'-2-4/j1 

Multiplication  of  Monomials 

121.  The  area  of  a  rectangle  and  the  surface  and  volume 
of  a  rectangular  block  can  be  easily  found  when  the  dimensions 
are  known. 

1.  Find  the  areas  of  rectangles,  whose  dimensions  are: 
(i)  5  in.  and  7  in.  (6)  m^  yd.  and  m  yd. 

(2)  13^  cm.  and  i6f  cm.  (7)  a'^h  mi.  and  h^c  mi. 

(3)  7  ft.  and  a  ft.  (8)  2j3  km.  and  T,y^  km. 

(4)  X  m.  and  9^  m.  (9)  5^:x;^;y  ft.  and  2>ixy^  ft. 

(5)  p  rd.  and  q  rd.  (10)  4 .  ^ahc  in.  and  3 .  2abc  in. 

Illustrate  the  problems  by  drawings  on  square-ruled  paper  when- 
ever possible,  and  check  results  by  counting  the  number  of  squares  con- 
tained in  each  rectangle. 

In  the  above  and  in  all  the  following  problems  with  literal  numbers 
test  the  values  obtained  by  assigning  numerical  values  to  the  letters. 

For  example,  the  area  of  a  rectangle  of  dimensions  2,db  and  25c  is 
6ah'c.  Assigning  to  a,  h,  and  c,  the  values  2,  3,  and  4  respectively,  the 
dimensions  and  area  are  18,  24,  and  432  respectively,  which  is  correct, 
since  18X24  =  432. 

2.  Find  the  area  of  a  square,  whose  edge  is:  15  ft.;  8i  m.; 
a^  cm.;  7.56c  in.;  x^  in.;  a^h"  mi.;  i^p'^q^  m.;  'jx^y^z  ft.; 
4.32a^6%4  dm.;   gmn'p^q^  cm. 

3.  Find  the  volume  and  area  of  a  cube,  whose  edge  is: 
7icm.;  2x' it.;  5^06  in.;  2.4y^m.;  ^xyz  cm.;  $.6x'y^z'  ml: 
3fa;4rd.;   i2a='6^  yd.;  4ias  ft.;  isp^q'r  m. 


Fundamental  Operations  and  Algebraic  Expressions    163 


4.  Find  the  area  and  volume  of  a  rectangular  block,  whose 
dimensions  are: 


(i)  4,  9,  and  15  in. 

(2)  3i  5i»  and  x  ft. 

(3)  ^.  8,  y  m. 

(4)  a^,  a3,  a4  rd. 

(5)  :x;3,  2X%  s:»5  yd. 


(6)  3-5/'^4-5/'^  i6/'Scm. 

(7)  jc^^z,  xy'z,  xyz^  ft. 

(8)  2amp,  ^bpm,  ^cmp  in. 

(9)  4f3e8,  2\x^°,  ^x^'  m. 
(10)  5^3^^,  4^x'y^,  ^-x'^y*  cm. 

5.  We  have  seen  (p.  21),  that  in  x^,  5  is  called  the  exponent, 
x  is  called  the  base,  and  x^  is  called  the  5th  power  of  x.  Give 
other  examples  of  different  powers  of  the  same  base. 

6.  In  what  way  are  powers  of  the  same  base  multiplied? 
Can  powers  of  different  bases  be  multiplied  in  the  same 
way? 


Fig.  195 


7.  What  is  the  volume  of  a  rectangular  block  of  dimensions 
a,  b,  and  c  ft.  ?  b,  c,  and  a  ft.  ?  b,  a,  and  c  ft.  ?  c,  b,  and  a  ft.  ? 
c,  a,  and  b  ft.  ?  a,  c,  and  b  ft.  ?  What  is  the  effect  of  the 
order  in  which  the  factors  are  multiplied  upon  the  value  of  the 
product  ? 

8.  What  is  the  total  volume  of  7  rectangular  blocks  of 
dimensions  x,  y,  and  z  ?  What  is  the  volume  of  one  rectan- 
gular block  of  dimensions  'jx,  y,  and  z  ?  x,  "jy,  and  z  ?  x,  y, 
and  7z  ?     "jx,  'jy,  and  z  ?     7:^,  )»,  7Z  ?    x,  'jy,  "jz  ?    7rx;,  yy,  yz  ? 


i64 


First-Year  Mathematics 


9.  Which  of  the  following  expressions  are  equal  ? 


(i)  ^{X'y^z) 

(5)  5(4 -3 -6)                 (9)8(7-6.5) 

(2)  'jx-iy-iz 

(6)  5.3-5.6.4            (io)8-7-8.6 

(3)  ix-yz 

(7)  5-4-3-6               (11)8.7.6.5 

(4)  IX'iyz 

(8)  5-3-5-4-5-6      (12)8.7.8.6 

8-5 


10.  Compare  also  the  following: 

5aX&XcXi,    5(aX6XcX</),    5aX5&X5cX5<^. 

11.  State  the  law  of  signs  for  multiplication  and  interpret 
the  meaning  of  positive  and  negative  multipliers  and  multi- 
plicands (see  pp.  77-85). 


Exercise  XXI 
Simplify  the  following  products: 

1.  (+i7)X(  +  25)X(-8) 

2.  (-a^)X(+fl4)X(-a3) 

3.  (-a)"X(-a)3X(-a)4 

4.  (2mr)(-3w*0(-5w3) 

5.  {-2\x^y){^-t;^x^yiz^{\\bxiyz'^) 

6.  6a^6c^X4a='6='cX3a^6'c='X5a6*c» 

+8.2a*m6)(— 3.56amn^6*)(+4.3w^n^6) 
-lxpqH){-2\xp^qr)i^-^%Pqr^) 

d,\lH^U^){-2\fS^U'^)i^-^t^°S''U^) 

i2x'y^z^     —^^x^y'z^    x^y^z' 
■  X  X  ■ 


9- 

10. 


13- 


14. 


5 

3 
+  7t 


X- 


6a3 


25 

24         14 

— a5*     105^ 
'7«r  X 
5         2 


15.  (a+6)3X(a+6)-« 


Fundamental  Operations  and  Algebraic  Expressions    165 

16.  (r"5)»X(r-5)3 

17.  (/^-/^)*X(/'-/*)* 

18.  (x^+y'yx{x^-{-y')^X{x'+y') 

19.  3{x+y)5{x--yyX4{x+yy{x-y)^ 

20.  \-{a-b)\iX\-^{a-b)\*X\-{a-b)\s 

i$x^y''{x-^y)^     —2ix'*y5{x+y)^ 

21.  /\ 

22.  (as)2X(-a3)4X(-a^)5 
23-  (3r)'*X(-/'3)sx(-s/.4)» 

24.    (2i^3)2X(-|:»S)2X(3^7)». 


Division  of  Monomials 

122.  We  have  found  the  area  of  a  rectangle  when  both 
dimensions  are  known.  We  shall  now  find  one  dimension  if 
the  other  dimension  and  the  area  are  known. 

1.  Find  the  altitude  of  a  rectangle  of  area  144  sq.  ft.,  the 
base  being  9  ft.;   16  ft.;  12  ft.;  8  ft.;  72  feet. 

2.  Denoting  the  area  of  a  rectangle  by  A,  the  base  by  6, 
the  altitude  by  a,  find  the  second  dimension  of  the  following 
rectangles: 

(i)  A=s^i'  sq-  ft. ;    b  =  26/  ft. 

(2)  i4  =5(;5  sq.  in. ;    6=:x;*  in. 

(3)  A=4^x^y3z^  sq.  m.;    a=;^x^y^z'  m. 

(4)  A=si.6d5f'c3;    a  =  2.4d3fc' 

(5)  A=ggp'*xn^;    b  =  iil'*x'p3 

(6)  A=gil^ksmcs;    b  =  i7,lk^mc' 

(7)  A=vsu^c'';    a=v'u^c' 

(8)  yl=3.5^%5z4;       J  =  5^2^304 

(9)  A=i. g6z3x5p(> ;     b  =  i.4Z''p''x' 
(10)  A=S^xSy^z^;    a  =  i^x3y'*zs. 

3.  State  a  short  way  of  dividing  powers  of  the  same  base. 

4.  State  the  law  of  signs  for  division  (see  p.  86). 


1 66 


First- Year  Mathematics 


2. 


4- 


6. 


lO. 


ExERasE  XXII 
Reduce  the  following  fractions  to  lowest  terms: 


—2$a^b^c'' 
Sa^b^cs 

—  i.6gx*ySz^wSa^ 
—I  .^a^'x'y^z* 

+  I2^/"m554 

i2{-aYbs{-c)^ 
3(-a)^63(_c)4 

—  2Sp^q5r3 

42ox'^y^^z3S 

—  i2X^^y''z^f 

56a*&*°c9 
I4a^67c6 

— 4.53f<'>'V 

6i/»''53»;'S'» 

i4(a+&)3 
-7(a+6)» 


II. 

12. 

13- 

14:. 

IS- 
i6. 

17- 


19 


20. 


-94(3c^->>^)5 

—  2(3(;*-3'^)3 

-3.43(a''  +  2a6+3')^ 
4g{a'  +  2ab+b'y 

7h-iP-q)\' 

S^x^y^ia' +b^y 
2^x^y3{a''-\-b')s 
-5\{a+b)3^(a-b)4* 

ilia+b)%{a-by'' 
—i44X''^y''''(x+yY+^ 

—iSxi<'y'*^{x+yy 
— i4f/>°gV(/>+g-r)°+^+*' 

6 


i.25a^&5c^s 


— 25a5&7c9 
4|(3g+y)3(jg->r)4 

-H(^+>')H^-)')'* 


Multiplication  and  Division  by  Means  of  Exponents 

123.  Make  tables  of  the  first  twelve  powers  of  2,  3,  and 
5;  thus: 

5=5^ 


2=2* 
4=2« 

8  =  23 
l6  =  2'» 
32=25 


3=3' 

9=3' 
27=33 

81=34 
243=3^ 


25  =  5" 

125=5^ 

625  =  54 

3,125=55 


Fundamental  Operations  and  Algebraic  Expressions    167 


64=26 

729  =  36 

15,625=56 

128=27 

2,187=3' 

78,125=5' 

256=2^ 

6,561=38 

390,625=58 

512=29 

19,683=39 

1,953,125  =  59 

I,024  =  2^<^ 

59,049=3'° 

9,765,625=5'° 

2,048  =  2" 

177.147=3" 

48,828,125=5" 

4,096  =  2" 

531,441=3" 

244,140,625=5" 

I .  Using  the  tables,  write  as  powers 

of  2,3,  015: 

16 

6,561 

177,147 

2,187 

243 

729 

1,024 

128 

15,625 

390>625 

48,828,125 

2.  Using  the  above  tables,  many  large  multiplications  and 
divisions  may  be  simplified,  as  will  be  seen  from  the  following 
examples: 

(1)  32X64  =  25X26  =  2"=2,048 

(2)  15,625  X625=56X5^  =  5'°=9,765,62S 

(3)  (729)' =  (3^)' =3'' =  531,441 

(4)  4,096-7-16  =  2"  ^24  =28  =  256 

(5)  (625)3  =  (54)3  =512  =244,140,625. 

Exercise  XXIII 

Carry  out  the  following  multiplications  and  divisions, 
using  the  tables  of  powers  of  2,  3,  and  5 : 

1.  6,561X81  7.  390,625^3,125 

2.  78,125X625  8.  244,140,625-4-78,125 

3.  512^16  9.  (243)=' 

4.  729X243  10.  (15,625)^ 

5.  2,048 -j- 1 28  II.  4,096-^64 

6.  177,1474-729  12.  390,625X625. 


i6. 
17- 


1 68  First- Year  Mathematics 

For  the  following  problems  the  above  tables  have  to  be 
extci\ded  to  the  25th  powers: 

282,429,536,481 
3,486,784,401 

14.  15,625X78,125X9,765,625 

15.  2563 

847,288,609,443X531,441 
31 ,381 ,059,609  X  14,348,907 

30,517,578,125X3,814,697,265,625 
10,670,928,955,078,125 
„    16,777,216X1,024X262,144 
2,097,152X8,192 

Multiplication  and  Division  of  a  Polynomial  by  a  Monomial 

124.  What  is  the  total  area  of  four  adjacent  flower-beds, 
whose  length  is  10  ft.  each,  and  whose  widths  are  7.  2,  5, 
and  3  ft.  respectively? 

The  total  area  may  be  expressed  in  either  of  the  two  following  ways : 

10(7  +  2  +  5  +  3)     or    10X7+10X2+10X5  +  10X3 
Thence: 

10(7 +  2 +  5 +  3)  =  10X7 +  10X2 +  10X5 +  10X3. 
The   area   may   thus    be  expressed    as    the  product  of  a  polynomial 
by  a  monomial  or  as  one  polynomial. 

1.  Represent  in  two  different  ways  the  total  area  of  7 
adjacent  rectangles  having  the  same  base  b  and  having  lengths 
5,  a,  X,  7 J,  4^,  9.5,  and  y'  respectively.  Express  by  an  equa- 
tion the  equality  of  these  two  representations. 

2.  Represent  in  two  different  ways  the  combined  area  of 
4  rectangles  of  length  a*  and  of  bases  a^,  T,ab^,  T,a'b,  b^  respec- 
tively.    Express  the  equality  of  the  two  representations. 

3.  How  may  the  product  of  a  polynomial  by  a  monomial 
be  reduced  to  a  polynomial  ?  What  do  you  notice  concerning 
the  terms  of  the  resulting  polynomial  ? 


Fundamental  Operations  and  Algebraic  Expressions     169 

Exercise  XXIV 
Reduce  to  one  polynomial: 

1.  Sx{x^--T,xy^sy'') 

2.  6§a&(3a^— 6^6  +  126^) 

3.  —$.']x^{\.lahx  —  j^,;^ahy~2^(^xyz-\-.^f^hyz) 

4.  —ap^m^{+4a'p^fn  —  ^a^pm'  —  'j^ap^m^—4.<,a^p^m) 

6.  5a='(2^a'  —4^(16—3 . 56^)  — 4&^(3|o^  +  2c6-2|6^) 

7 .  ^^ {;^x  —  2y)  —  2xy{T,x  —  2^)  +  6 j^ {7,x  —  23^) . 

8.  2(3a^-4io6  +  7&') -3(40^  +  5^-8^6*)  + 

4K-2|o^-6fl&-5fe^) 

9.  S5(;|4a-2(3a-46)+5(4a-36)| 

10.  2a|5(4a  — 76  — 3c)— 6(5a  +  46— 8c)[ 

11.  — 4Jc[2:x;^  +  33£;|4(:x;  — i)  — 5(51;  — 2)^] 

12.  +5)'-^-[3>'^-23'^4>'(>'  +  3)-5>'(2>'+6)n- 

Solve  the  following  equations: 

13.  4(^-6) -3(:x:  +  3) -15 

14-  3i(3^-8)-4K3^  +  39)=o 

15.  —  ^x{4x-\-  2y  +  6)  +  T,x{'jx  -\-6y —g)  —  {x'  +  Sxy  —  jx) 

=  100 

16.  4a[3a-(5a  +  2)]-2a[-3a-(a-4)]-4(-3a-5)=o. 

125.  To  factor  a  number  is  to  find  the  numbers  which, 
multiplied,  give  the  number  to  be  factored. 

1.  The  total  area  of  three  adjacent  lots  is  x'*  +  2i^^y+4x^y^ 
sq.  m.,  each  term  representing  the  area  of  a  lot.  The  lots 
all  have  the  same  length.     What  may  their  dimensions  be  ? 

2.  Sketch  a  rectangle  whose  area  is: 

(i)  4X^-3xy 

(2)  ^x^  —  iox'y  +  i$xy^ 

(3)  i4a^b3c^—2ia3b^c^  +  T,sa^b^c3 

(4)  3w5  — 12 w^n+Omw* 

(5)  i^x'^  —  iox^  +  s^'. 


i7o 


First-Year  Mathematics 


3.  How  can  some  polynomials  be  reduced  to  the  product 
of  a  polynomial  by  a  monomial?  Can  every  polynomial  be 
factored  in  this  way  ? 

Exercise  XXV 

Write  the  following  polynomials  as  the  product  of  a  poly- 
nomial by  a  monomial: 

1.  5a  — 106 

2.  i'jx'  —  2Sgx3 

3.  i6x'  —  2abx 

4.  ax-{-ay—az 

5.  i^x^'y'z^  —  'jx^y^z^+Sxy'z'' 

6.  6om'n^r'  —^pn^n^r^  +gom'^n^r^  +()om^n^r' 

7.  4.5a»63c4_i.2a36*c'»  — 2.7a'»6^c3+9a364c» 

8.  4a'x3  —  i2a^x^  —  2oa'^x3 

9.  ap3-\-^p'x-4pq  +  isapxq 
10.  x^—pqx'+p'q^x—pq^x^. 

126.  When  one  of  the  factors  of  a  number  is  known,  a 
second  factor  is  found  by  dividing  the  known  factor  into  the 
given  number. 

1.  The  area  of  a  garden  is  i^abc-{-2']a'c—2ib'c.  The 
length  is  3c.     Find  the  width. 

2.  Find  the  missing  dimension  of  the  following  rectangles; 

AREA  BASE  ALTITUDE 

(1)  54+57  5"  — 

(2)  8x^y'  —  ioxy3—4x*  —  2X 

(3)  pk+lk-rk+tk  -  k 

(4)  mu'+mv^+mw'^—mz  m  — 

(5)  as+bs—cs+ds  s  — 

3.  How  may  the  quotient  of  a  polynomial  by  a  monomial 
frequently  be  reduced  to  a  polynomial  ?  Is  this  always  pos- 
sible ? 


Fundamental  Operations  and  Algebraic  Expressions    171 

Exercise  XXVI 
Reduce: 

2oa^h  —  1 5a'&^  +3og&^ 
506 

I  ox^y — I  s^^y^ + s^^y 

-Sxy 
2)6x^y^—/^2x^y^z 


—6x'y 

35a6*c3  — 42a3&4c3  _4Qa8j4c3 -|-2ia36^c' 
7a&3c3 

i2am'n^—i6bm^n'+4oabm'n'  —  2Sa'm'n' 
—  4mn 

sri'  ■ 

4abc{^abc —$a^b^c'  —  'jab^c+ 6ab^c) 

—  2a''bc^ 

2x{/^yz—6x''y'')—%y^{$xz^  —  i\x^) 

^xyz 

i^a^b'^cs—ga'^b^c'  —^^a'b^cs  +  2ia^b'*c3  —^a^^b^c* 

^a^'b^c^ 

2 1  abdpqs  —  T,^abcpqt — 42  acdpts 

10. 7 •. 

-lap 

Exercise  XXVII 
Reduce  to  a  polynomial: 

1.  A^abci^a'-ib'+^c') 

2.  ;^1iX'y3{x3y'*—^'^y^ ^-6x'y'^  —gx^y^) 

3.  — 5a[4a&c(3a^&— 5a&'+3a='c— 6ac^)-3a^&c2(4a— 5&-6c).] 

Reduce  to  the  product  of  a  polynomial  by  a  monomial 
or  by  a  binomial: 

4.  25a^b'^c''d^ —isa^b^cH''  ^^oab^c^d^ 

5.  2ix^y^z^^  —  i4X^yz^'^-{-^$xy^''z^'—42X^''y^H 

6.  4a{x-y)-zb{x--y)+4c{x-y). 


172 


First-  Year  Mathematics 


Reduce  to  a  polynomial: 

I  ^x'<^y'''z''^  —  I  Sx^'^y^h^'^  —  21  x^''y'^''z^ 
'■  ^x^y^z^ 

g    i5(x+y)^--25{x+y)^-SSix+yy 

-si^+yy 

[21  pm{r'+s'){r+s)-42mp{r'+s'y- 7  m]  (r'+s'){r^-sy 
^'  -7m(r='+5=')(r+5) 

Solve  the  following  equations  for  c: 

10.  4c(3f-3y+4Z-2)-4c(4Z-5)+3c(4>'-2)-c(4X-i) 

=35 

11.  sc[4r='-3c(c+2)]-s|c3-6c2-c[=2o 

12.  3a(c— 46)  — 26(4c— 6a)— 4c(a  — 26)=ioa. 

Multiplication  of  Polynomials 

127.  When  the  dimensions  of  a  rectangle  are  polynomials, 
the  area  can  be  found,  by  separating  the  rectangle  into  other 
rectangles  whose  dimensions  are  simpler  numbers. 

1.  Draw  a  rectangle  of  length  x-\-y  and  of  width  a-\-b. 
Determine  the  area  of  the  rectangle  and  write  your  answer 
in  the  form  of  a  polynomial. 

2.  Express  the  area  of  the  rectangle  of  Fig.  196  (i)  as  the 
sum  of  two  rectangles;   (2)  as  the  sum  of  four  rectangles. 


c  a 


Fig.  196 


c  -?■ 

Fig.  197 


3.  Express  the  area  of  the  rectangle  of  Fig.  197  (i)  as  the 
sum  of  two  rectangles;   (2)  as  the  sum  of  four  rectangles. 

4.  Write  the  area  of  the  rectangles,  whose  dimensions  are 
given  below,  (i)  as  the  sum  of  two  or  more  rectangles,  (2)  in 
the  form  of  a  polynomial.     Simplify  the  polynomial: 


Fundamental  Operations  and  Algebraic  Expressions     173 


LENGTH 

WIDTH 

(i)  3+4 

6  +  2 

(2)  s+a 

<J  +  3 

(3)  «+6 

a  — 2 

(4)  a-\-b 

b+c  . 

(5)  »+:y 

y-\-z 

(6)  a''+b'  +  2ab 

a+b 

(7)  x''  +  2xy-\-y' 

x—y 

(8)  a-b 

a''  —  2ab-\-b' 

(9)  ^+3' 

x^  —  2xy+y' 

(10)  s+x 

25  +  io:x;+:x;^. 

5.  From  the  preceding  problems  make  a  rule  for  express- 
ing the  product  of  two  polynomials  in  the  form  of  a  single 
polynomial. 

128.  The  rectangles  of  problems  2,  3,  and  4,  §127,  are  first 
expressed  as  the  sum  of  two  or  more  rectangles;  for  example, 
for  the  rectangle  of  problem  2 : 

{a-\-b){c  +d)  =a{c-\-d)  +b{c+d). 

The  final  product  of  the  two  polynomials,  which  represent 
the  dimensions  of  the  rectangle,  is  thus  made  up  of  the  sum 
of  products  of  one  of  the  polynomials  by  the  several  terms  of 
the  other  polynomial.  These  separate  products  are  called 
the  partial  products.  They  represent  what  each  term  of  one 
of  the  polynomials  in  connection  with  the  entire  second  poly- 
nomial contributes  to  the  final  product. 

1 .  State  the  partial  products  for  each  of  the  parts  of  prob- 
lem 4,  also  stating  how  each  of  these  is  obtained. 

2.  Most  frequently  the  partial  products  are  written,  one 
underneath  the  other,  so  as  to  bring  all  similar  terms  in  the 
same  column.     The  work  is  then  arranged  as  follows: 

a^  +  ab  +  b^ 
a  +b        _ 
First  partial  product:        a(a'  +  ab  +  b''):       a3+  a^b+  ab' 
Second  partial  product:     b{a^  +  ab -\-b^):  a'b+  ah^  +  b3 

a3  +  2a'b  +  2ab''  +  b3 


174  First-Year  Mathetnatics 

Exercise  XXVIU 

Simplify  the  following  products: 

1.  {ax-\-by){ax—by) 

2.  (a3+a6='+&3)(a+&) 

3.  {p'  +  2p  +  i){p-3) 

4.  (2ia*-3J63)(i2a3-66^y 

5.  {4.6abc+i.2ab^){^a'c—sa'b) 

6.  {a'+ab+b'){a-b){a+b) 

7.  {4X''  +  2xy+y'){2X+y){2X—y) 

8.  {p'+pri-r'){p'-pr+r') 

10.  (2jm^-4^w5+3i5^)(Vm— V^) 

11.  (306— 56c)(2a— 36  +  5c)  +  (4^— 6a6)(3c2— 46*  +  2a*) 

12.  {gx'—6xy+4y'){gx'+6xy+y') 

13.  (5a +26)3 -(5a -26)3 

14.  (2/-3/t)^-(2/+3/^)'  +  (2/-3fe)(2/+3^)- 

129.  The  squares  of  binomials  can  be  deduced  readily 
from  the  binomials  without  actually  performing  the  multipli- 
cations. 

1.  Express  in  the  form  of  a  trinomial  by  means  of  multipU- 
cation: 

(m-xy  (/»+*)» 

ia+by  {x-yY 

{p-ky  (w+«)'. 

Notice  how  the  terms  of  the  binomial  that  is  squared  are 
used  to  form  the  terms  of  the  final  result. 

2.  Write  down  at  once  in  the  form  of  a  trinomial: 

{x^-py  {k-ry  ir-yy 

{v+ay  ir->ryy  {h-py 

{x-py  {k+ny  {g-py. 

3.  From  the  preceding  problems  make  a  rule  for  expressing 
the  square  of  a  binomial. 


Fundamental  Operations  and  Algebraic  Expressions    175 


ExERasE  XXIX 

Write  in  the  form  of  a  trinomial,  doing  all  you  can  men- 
tally: 


7.  {.6xyz-\-iY 

8.  [(a+&)+3? 

9.  [{a-h)+cY 

10.  [{x-y)-^Y 

11.  [(^+/)-i.5^? 

12.  [(3a-46)-2Cp 


1.  (^  +  5)^ 

2.  Cv-7)^ 

3.  (4a-i)* 
4-  {hx-V^yY 

5.  (.4a-. 3/)^ 

6.  (3px-4^>')^ 

13.  [{ax-\-hy)—czY 

14-  [(5^7-3^0 -4/'/]^ 

15.  [(2^a6^c— fa6c=')+7a2&c]* 

16.  [(a+6)  +  (c-(/)]^ 

17.  (2a+36)^-(3a-2&)^  +  (4a-3&)» 

18.  [(4^-5^)  +  (5^-4>')]^ 

130.  The  product  of  the  sum  of  two  numbers  by  the  dif- 
ference of  the  same  numbers  can  be  readily  obtained  without 
actually  multiplying. 

I.  Express  as  a  binomial  by  multiplying: 


{a-\-x){a—x) 

(r+4)(r-4) 
{t^q){t-q) 

2.  Write    in    the    form 
plying: 

{k^h){k-h) 
(25 -1 3)  (25 -hi  3) 
{in-\-v){ni—v) 
{t-u){t-\-u) 


(p+s){p-s) 
(y-\-b)\y-b) 
(i5-z)(i5-}-z). 

of    a    binomial    without    multi- 


(4a -36)  (4a +36) 

( .  3^/  +  •  2yk)  ( .  33c;/  — .  2yk) 
{abc —xyz)  {abc+xyz) . 


3.  From  the  preceding  exercises  make  a  rule  for  expressing 
as  a  binomial  the  product  of  the  sum  of  two  numbers  by  the 
difference  of  the  same  numbers. 


176  First-Year  Mathematics 

Exercise  XXX 

Express  as  binomials  and  perform  the  operations  indicated 
in  the  resulting  binomials: 

1.  (2a +36) (2a -36) 

2.  {7,x-^z){t,x-\-^z) 

3-  (5/'+ 3^)  (5/' -3^) 

4-  ihr^¥){hr-¥) 

5-  {lxy-lxz){^lxz^-lxy) 

6.  (4ak-3)(4a&c  +  3) 

7.  (i+5/'T)(i-5r?') 

8.  (i-a)(i+a)(i+aO 

9.  (m— •y)(M+z')(M^+^^^) 

10.  [(a+&)+c][(a  +  6)-c] 

11.  [(e-6)-;j][(e-6)+;i] 

12.  [(w+w)-/^][(w +«)+/*] 

13.  [(2a-&)-35][(2(i-6)+35] 

14.  (/fe-0()fe+/)()fe^+/^) 

IS-  (^-/)(^+/)(r+/^)(r+/^) 

16.  (i -/,)(! +^)(i+^a) (1+^4) 

1 7 .  [(3:x;'» — 2^3)  —xy]  [{^x'^  --2yi)-\- xy] 

19.  (a'»+&4)(a'»— 64) 

20.  {x<^—y^){x^-\-y'') 

21.  (2m*'  — 353'')(2W2'  +  353'') 

22.  [(a+6)-(c+i)][(a+6)  +  (c+(/)] 

23.  [{X^  -y^)  -  (^3  +^3)]  [(^2  _^a)  _,.  (^3  +3,3)] 

24.  [(i  .2a6  +  J&c)  — (.4ac-26^)][(i.2a&+4&c)  +  (.4ac-26=')] 
25-  [(i^^  -h')  +  (f ^^  +i)'^)]  [(i:v^  -§y^)  -  {^x^+h')l 

Exercise  XXXI 
Multiply  as  indicated: 

1.  2a^(3a3— 463) —3^3(40^  — 2&^) 

2.  {\pt-\ts){\pt-%ts) 


Fundamental  Operations  and  Algebraic  Expressions    177 

3.  {a+h^cY 

4.  {a-\-h~2cy 

5.  (-2a-36+c)" 

6.  (fl» +6^-3^3)2 

7.  (2a^+6^  +  3C^)(2a^  +  62-3c^) 

8.  (.5:x;— .4^'— .3z)(io.\:  — 20}'+30z) 

9.  4iww(4w^— 6wn+9n*)— 3w;j^(i3i»— 9m) 

10.  (3>''-45^)'+6^')(3>''-5^') 

11.  (a:x;+&>'— cz)(a5(;+6)'4-cz) 

12.  {ax-\-hy—cz)'^ 

13.  (a:x;— &)'— cz)' 

14.  (a:c— 6y+cz)' 

15.  {\ms—\st—^tm){6m  —  i2S-^i2>t) 

16.  (—1 .4/*  — 2.5W/+  .gm'){.2l^—lm—  .iw*) 

17.  (a^— 6y+cz)^  — (ajv+^y+cz)^ 

18.  {a+by 

19.  (a— by 

20.  (25c— )')3 

21.  (»+2j)3 

22.  (3^+4>')^-(3^-4>')3 

23-  (iw^-W^  +  (im^  +  M3 

24.  (.5^--6>')'-(-5^+-6:y)'-(-S^+-6>')(-5^--6>') 

25.  (2/>'+9^)3-(2/)3_g3)2_4^4(p_2g)^ 


Division  of  Polynomials 

131.  In  the  division  of  polynomials,  all  the  preceding 
operations  find  application.  It  is  therefore  a  subject  by  means 
of  which  the  foregoing  work  may  be  reviewed. 

I.  Write  the  product  of  x+y  by  a+b-\-c  in  the  form  of  a 
polynomial. 

What  is  the  contribution  of  a  and  a£;+;y  to  the  product? 
Of6and5c+y?     Oi  c  and  x+y? 


1^8  First-Year  Matliematics 

From  what  terms  of  the  multiplier  and  the  multiplicand  is 
the  term  ax  of  the  product  obtained  ? 

2.  What  is  to  be  found  by  dividing  ax-\-ay+bx+by-^cx  + 
cy  by  x+y? 

How  is  the  first  term  of  the  quotient  found  ? 

How  is  the  total  contribution  of  the  divisor  and  the  first 
term  of  the  quotient  obtained  ? 

Write  down  the  remainder  of  the  dividend,  which  is  con- 
tributed by  the  divisor  and  the  remaining  terms  of  the  quotient, 
as  yet  unknown. 

3.  Determine  the  2d  term  of  the  quotient  from  the  re- 
mainder of  the  dividend  by  the  same  method  as  was  used  in 
problem  2  to  determine  the  first  term  of  the  quotient.  Then 
proceed  as  in  problem  2.  Repeat  this  process,  until  all  the 
terms  of  the  dividend  have  been  used. 

The  following  arrangement  of  the  work  is  convenient: 

a+b+c 
x+y/ax+ay  +  bx+by+cx+cy 
First  partial  product:  a(x+y)     ax  +  ay 


bx  +  by  +  cx+cy 
Second  partial  product:  b{x+y)  bx  +  by 


cx+cy 
Third  partial  product:  c{x+y)  cx+cy 


4.  Compare  the  partial  products  of  problem  i  at  the 
beginning  of  this  section  with  the  partial  products  in  the 
division. 

5.  Divide  a^+^a'b+^ab'+b^  by  a-\-b.  What  is  the  first 
term  of  the  quotient?  How  is  the  ist  partial  product  found? 
What  is  the  2d  term  of  the  quotient  ?  How  is  the  2d  partial 
product  found  ?  etc. 


Fundamental  Operations  and  Algebraic  Expressions    179 

Exercise  XXXII 
Divide,  and  test  by  multiplying:* 

1.  9^^  +  245^+165^  by  3/ +45 

2.  1 .2X^+ .$xy  —  2.Sy^  hy  ^x  +  yy  X 

3.  6k^—^ikH+4'jkt''—42t^  by  2k  — yt 
*4.  iox3—$xy^—^^y3—^^y  by  x  —  2y 

5.  27a3  — 54a^6  +  36a&^— 863  by  3a  — 26 

6.  27a3  — 54a='&+36a62— 863  by  ga^"  — i2a&+46« 

7.  a;^— 3>3  by  x—y 

8.  27/3—6453  by  3^—45 

9.  64a^— 6^  by  8a3— 63 

*io.  ^y* ■\-i$x3y—gxy3  —  25X* -{-lox'y*  by  5a;*— 3^^ 

11.  8:>i;3— ;y3  by  4X^  +  2xy+y' 

12.  64a^— 6^  by  4a='— 6^ 

13.  . 00853/3 +'y3  by   .2St  +  V 

14.  a3  4-a26+a6^+ac2+6c*+63  by  a+6 

15.  64a^— 6^  by  i6a'*+4a''b''+b'* 

1 6.  /^X^  + 1 2;x;3;y3  +  gy^  by  2X3  4.  ^^3 

17.  . 00853/3 +t;3  by  .o45='/»—. 25/7;+^' 

18.  le^x:^  — 25^4  by  45(;*  — 5^2 

19.  a3+a^6+a62+ac*+6c^+63  by  a^+6='+c" 

20.  9/^  — 12/353+456  by  3/3— 253 

21.  27w<^— 8»*^  by  9W»+6w^«*+4»4 

22.  i6rx;4  — 25r4  by  4:x;^  +  5r"  — '. 

23.  2Tm^+&n^  by  T,m'  +  2n' 

24.  64w6  +  i4.4M'*z'^  +  i  .o8w^7;'»+.o27t/*^  by  4M^  +  .37; 

25.  2'jm^—Sn^  by  3w='  — 2»^. 

*  Before  beginning  to  divide  arrange  the  dividend  and  the  divisor 
according  to  powers  of  the  same  number.  Arrange  the  successive  re- 
mainders in  the  same  order.  Although  the  work  can  sometimes  be 
done  just  as  well  without  paying  any  attention  to  the  arrangement,  it 
will  be  found  convenient  to  follow  the  above  rule. 


i8o  First- Year  Mathematics 

Summary 

Terms  which  have  a  common  factor  are  called  similar  with 
respect  to  that  factor. 

Terms  which  have  no  common  factor  are  called  dissimilar. 

Polynomials  consisting  of  similar  terms  may  be  simplified  by 
adding  the  coefficients  of  the  common  factors  in  the  similar  terms. 

Polynomials,  no  two  of  whose  terms  are  similar,  cannot 
be  simplified. 

Subtraction  of  monomials  and  of  polynomials  may  always 
be  redticed  to  addition. 

If  a  polynomial  is  in  parentheses  preceded  by  a  minus 
sign,  the  subtraction  indicated  may  be  changed  into  addition 
by  changing  the  sign  of  each  term  in  the  parentheses. 

The  value  of  a  product  remains  unchanged,  when  the 
order  of  the  factors  is  changed. 

A  number,  which  is  expressed  as  the  product  of  two  or 
more  factors,  is  multiplied  by  multiplying  one  factor. 

Powers  of  the  same  base  are  multiplied  by  adding  the 
exponents  of  the  powers. 

Two  powers  of  the  same  base  are  divided  by  subtracting 
the  exponent  of  the  divisor  from  the  exponent  of  the  dividend. 

Multiplication  and  division  of  arithmetical  numbers  may 
be  simplified  by  the  use  of  exponents. 

To  multiply  a  polynomial  by  a  monomial,'  multiply  each 
term  of  the  polynomial  by  the  monomial  and  ctdd  the  separate 
products  algebraically. 

To  divide  a  polynomial  by  a  monomial,  divide  each  term 
of  the  polynomial  by  the  monomial,  then  add  the  separate 
quotients  algebraically. 

The  product  of  two  polynomials  is  obtained  by  multiplying 
one  of  the  polynomials  by  every  term  of  the  other  polynomial, 
and  adding  the  partial  products  algebraically. 

The  process  of  dividing  one  polynomial  by  another  is  the 
reverse,  step  by  step,  of  the  process  of  multiplying  polynomials 


CHAPTER  IX 

PRACTICE  IN  ALGEBRAIC  LANGUAGE.     GENERAL 
ARITHMETIC 

Translating  Verbal  into  Symbolic  Language 

132.  The  following  exercises  give  practice  in  translating 
number  expressions  and  relations  from  verbal  into  symbolical 
language  and  the  reverse.     Ansv^^er  all  you  can  orally. 

1.  Give  the  three  consecutive  integers  that  begin  with  16; 
end  with  16. 

2.  Express  the  three  consecutive  integers  that  begin  with 
»;  end  with  n. 

3.  Express  the  three  consecutive  even  numbers  that  begin 
with  2x;  end  with  2X. 

4.  Express  the  three  consecutive  odd  numbers  that  begin 
with  «+3;  end  with  w  +  3. 

5.  Express  the  sum  of  the  three  integers  that  begin  with  x 
and  differ  by  3. 

6.  Express  the  sum  of  the  three  consecutive  integers,  the 
second  integer  being  n. 

7.  Express  in  symbols  that  the  sum  of  three  consecutive 
integers  equals  21.     Find  the  three  integers. 

8.  Express  in  sjmabols  that  the  product  of  two  consecutive 
integers  equals  72. 

9.  Express  in  symbols  that  the  sum  of  three  consecutive 
integers,  differing  by  3,  equals  27.     Find  the  three  integers. 

10.  The  tens'  digit  of  a  certain  number  is  4  and  the  units' 
digit  is  3.  Indicate  in  symbols  the  number  of  units  in  the 
tens. 

11.  The  tens'  digit  of  a  number  is  t  and  the  units'  digit  is  u. 
Indicate  the  number  of  units  in  the  number. 

181 


i82  First-Year  Mathematics 

12.  The  tens'  digit  of  a  number  is  twice  the  units'  digit 
Indicate  the  number  in  symbols. 

13.  The  present  age,  in  years,  of  a  person  is  denoted  by  x 
Indicate  in  symbols  the  following: 

(i)  The  person's  age  12  years  ago 

(2)  His  age  12  years  hence 

(3)  His  age  when  twice  as  old 

(4)  Four  times  his  age  8  years  ago 

(5)  Six  times  his  age  3  years  hence 

(6)  One-eighth  of  his  age  15  years  ago 

(7)  Three  times  his  age  less  twice  his  age  6  years  ago 

(8)  60  decreased  by  his  age 

(9)  His  age  decreased  by  60 

(10)  His  age  increased  by  one-half  his  age 

(11)  His  age  diminished  by  3  times  his  age  20  years  ago 

(12)  The  ratio  of  his  present  age  to  his  age  14  years 

hence. 

14.  Letting  a  certain  number  be  denoted  by  10/+M,  what 
will  denote  the  number  formed  by  reversing  the  digits  ? 

15.  \i  X  denotes  the  number  of  minute-spaces  the  tip  of  the 
hour  hand  moves,  over  in  a  given  time,  what  will  denote  the 
number  of  minute-spaces  the  tip  of  the  minute  hand  moves 
over  in  the  same  time  ? 

16.  If  X  minutes  denote  the  space  moved  by  the  minute 
hand,  what  denotes  the  space  moved  by  the  hour  hand  in  the 
same  time  ? 

17.  Express  in  symbols  that  a  number  whose  units'  digit 
is  3  less  than  the  tens'  digit  equals  27  more  than  the  number 
made  by  writing  the  digits  in  reverse  order. 

18.  Express  that  one-eighth  of  the  double  of  n  increased 
by  25,  equals  one-half  of  ». 


Practice  in  Algebraic  Language  183 

19.  Write  in  symbols  an  expression  for  double  a  number, 
increased  by  3  times  the  sum  of  the  number  and  4. 

20.  Write  an  expression  for  double  a  number,  decreased 
by  3  times  the  difference  between  the  number  and  4. 

In  the  phrases  "  the  difference  between  8  and  4,"  "  the  difference 
of  8  and  4"  and  the  like,  it  is  understood  that  the  first-mentioned  num- 
ber (the  8)  is  the  minuend;    thus,  8  —  4,  not  4—8. 

21.  Write  the  double  of  a  munber,  decreased  by  3  times 
the  difference  of  the  number  and  2. 

22.  Write  in  symbols  4  times  the  difference  of  a  number 
and  3,  decreased  by  3  times  the  difference  of  the  number  and  i. 

23.  Show,  in  sjnnbols,  that  the  double  of  a  number, 
increased  by  3  times  the  difference  of  the  number  and  4, 
equals  13. 

24.  Find  the  value  of  the  following  for  :x;  =  i2: 

2X  +  7,{x-^4)  2X-^(x+4) 

2X  +  S{X-^)  2X-s{x-4) 

Six-4)-2X  4{x-s)-3{x-6). 

25.  Translate  the  six  expressions  of  problem  24  into  words. 

26.  Write  in  symbols  the  following: 
(i)  One-third  the  sum  of  r  and  4 

(2)  The  sum  of  r  and  4,  divided  by  3 

(3)  One-half  the  difference  of  twice  5  and  11 

(4)  The  difference  of  twice  s  and  11,  divided  by  2 

(5)  Two-thirds  of  the  difference  of  Sy  and  20 

(6)  Two  times  the  difference  of  Sy  and  20,  divided  by  3 

(7)  One-fourth  of  5  times  the  sum  of  2X  and  9 

(8)  Five-fourths  of  the  sum  of  2X  and  9. 

27.  Show,  in  symbols,  a  number  whose  digits  are  x  and  4; 
X  and  y ;  y  and  x. 

28.  Show,  in  symbols,  a  three-digit  number  whose  digits 
are  a,  b,  and  c;  c,  a,  and  b;  x,  z,  and  y,  y,  z,  and  x. 


184  Fir  si- Year  MathenuUics 

The  Operations  of  Arithmetic  Symbolized 

133.  The  work  of  algebra  requires  that  the  operations 
used  to  obtain  results  be  indicated  more  frequently  and  more 
fully  than  they  are  in  arithmetic. 

1.  Show  by  an  equation  that  12  is  the  sum  of  7  and  5. 

2.  Indicate  that  45  is  the  sum  of  4  tens  and  5. 

3.  Indicate  that  68  is  the  sum  of  6  tens  and  8. 

4.  Indicate  the  sum  of  50  and  7;  48  and  9;   18  and  x. 

5.  What  is  the  svun  of  8 and x?  Of  1 5  and y?  Oix and  16 ? 
Of  X  and  y  ? 

6.  Show  by  an  equation  that  s  is  the  sum  of  a,  a,  a,  a, 
and  10. 

7.  The  minuend  is  18  and  the  subtrahend  7.  Express  by 
an  equation  that  the  difference  is  11. 

8.  Denote  the  minuend  by  m,  the  subtrahend  by  s,  and 
the  difference  by  d.  Express  by  an  equation  the  relation 
of  these  numbers. 

Equation:  d=m—s.  Interpretation:  "  difference  equals  minu- 
snd  minus  subtrahend. 

9.  Express  by  an  equation  that  3  times  a  equals  ^x. 

10.  The  multiplicand  is  M,  the  multiplier  m,  and  the 
product  P.  Express  their  relation  by  an  equation  and  trans- 
late the  equation  into  words. 

11.  Add  s  to  both  sides  of  the  equation,  d=m—s,  and 
translate  the  result.  Show  that  the  result  is  a  rule  for  check- 
ing, or  testing,  subtraction. 

12.  Divide  both  sides  of  P=M-m,  or  P=Mm,  (i)  by  w 
(w-minor)  and  interpret  (translate  into  words)  the  result;  (2) 
by  M  (M-major),  and  interpret  the  result. 

13.  Show  by  an  equation  the  relation  between  the  dividend, 
D  (D-major),  divisor,  d  (i-minor),  and  the  quotient,  q,  when 
there  is  no  remainder.     Interpret  the  equation. 

14.  Show  by  an  equation  the  relation  of  the  dividend  D 


Practice  in  Algebraic  Language  185 

divisor,  d,  quotient  q,  and  the  remainder  r,  and  interpret  the 
equation. 

Express  by  equations  the  relations  of  the  numbers  in  the 
following  problems: 

15.  A  boy  has  m  marbles  and  buys  h  more.     He  then  has 
M  marbles. 

16.  There  are  h  boys  and  g  girls  in  a  class  of  p  pupils. 

17.  A  boy  earns  c  cents  a  day  for  d  days.     He  then  has 
5  cents. 

18.  A  rectangular  flower  bed  is  /  feet  long  and  contains 
s  square  feet.    The  width  is  w  feet. 

19.  A  bicyclist  rides  M  miles,  which  is  r  miles  more  than 
t  times  h  miles. 

FRACTIONS 

20.  The  sum  of  the  fractions  -^  and  -^\s  S. 

di  dj 

fi  ft 

21.  The  difference  of  the  fractions  —  and  -^  is  D.     (^  is 


dj  d 

the  minuend-fraction.) 

ft  ft 

22.  The  product  of  the  fractions  -y-  and  -^  is  P. 

dj  0,2 

ft  ft 

23.  The  quotient  of  the  fraction  ~  divided  by  -^  is  Q. 

MENSURATION 

24.  The  base  of  a  rectangle  is  b  ft.  and  the  altitude  is 
a  ft.    The  area  is  R  sq.  feet. 

25.  Each  side  of  a  square  is  5  ft.  and  the  area  is  A  sq.  feet. 

26.  The  value  of  a  load  of  corn  of  b  bu.  at  5  ct.  per  bushel 
is  D  dollars. 

DECIMAL  FRACTIONS 

27.  A  decimal  fraction  has  three  units  in  tenths'  place, 
and  5  units  in  hundredths'  place,  and  the  value  is  v. 

Note.    •y  =  i\  +  T8o. 


1 86  First-Year  Mathematics 

28.  A  decimal  fraction  has  /  units  in  tenths'  place  and  h 
units  in  hundredths'  place  and  the  value  is  v. 

29.  A  decimal  has  a  units  in  tenths'  place  and  h  units  in 
hundredths'  place  and  c  units  in  thousandths'  place.  The 
value  is  v. 

30.  A  mixed  number  has  a^  units  in  hundredths'  place, 
tta  units  in  tenths'  place,  a^  in  units'  place,  a^  in  tens'  place, 
and  a;  in  hundreds'  place.    The  value  is  v. 

31.  A  mixed  number  has  a,  units  in  hundredths'  place 
and  a,  units  in  hundreds'  place.    The  value  is  v. 

PERCENTAGE 

32.  Find  6%  of  $210;  of  520  acres;  of  650  miles;  of  6 
yards. 

33.  Find  the  percentage,  p,  of  $450  at  5%;  of  375  bushels 
at  20%;  of  1,800  men  at  15%;  of  h  acres  at  25%;  of  b  at  r%. 

These  problems  show  that  the  law  of  percentage  may  be 

,  b'r  br 

written  thus:  p= —  ,  or  p= . 

100  100 

b-r 

34.  Divide  both  sides  of  the  equation  p= —    by  b   and 

100 

interpret  the  resulting  equation. 

b.r 

35.  Multiply  both  sides  of  p= by  100,  then  divide 

100 

both  sides  of  the  resulting  equation  by  b,  and  interpret  the 
result. 

36.  Given  the  percentage,  28.80,  and  the  base,  360,  find 
r,  the  rate.     Use  the  result  of  problem  35. 

37.  Given  the  percentage,  30,  and  the  base,  725,  find  the 
rate,  r. 

b-r 

38.  Multiply  both  sides  of  p= by  100  and  divide  both 

100 

sides  of  the  resulting  equation  by  r,  and  interpret  the  result. 

39.  Given  the  percentage,  27,  and  the  rate,  5,  find  the 
base,  6. 


Practice  in  Algebraic  Language  187 

40.  Given  the  percentage,  31.5,  and  the  rate,  7,  find  the 
base,  h. 

b-r 

41.  Multiply  both  sides  of  p= by  100  and  interpret. 

42.  The  percentage  is  77.4.  Find  the  product  of  the  base 
and  rate. 

SIMPLE  INTEREST 

43.  Find  the  simple  interest  at  6%  on  $65  for  i  year;  for 
2  years;  for  3  years;  for  5  years;  for  2^  years;  for  3I  years; 
for  t  years. 

44.  Find  the  simple  interest  for  t  years  on  $80  at  4%;  at 
5%;  at  6%;   at  4^%;   at  5.6%;   at  r%. 

45.  Find  the  simple  interest  at  r%  for  t  years  on  $20;  on 
$75 ;   on  $135;   on  $p. 

46.  Calling  i  the  simple  interest  on  a  principal  of  $p,  at 
r%,  for  /  years,  show  by  an  equation  how  i,  p,  r,  and  /  are 
related. 

pri 

47.  Divide  both  sides  of  the  equation,  i  =  ^-—  ,  by  rt,  then 

100 

multiply  both  sides  of  the  resulting  equation  by  100,  and  inter- 
pret. 

prt 

48.  Divide  both  sides  of  i=- —  by  pr,  then  multiply  both 

100  "^ 

sides  by  100,  and  interpret. 

prt 

49.  Multiply  both  sides  of  i=-- —  by  100,  then  divide  by 

pt,  and  interpret. 

What  is  the  product  of  a  fraction  by  the  denominator  ?      Illustrate. 

134.  Equations,  such  as  i=m— 5,  P=m-M,  D=q  •  d-\-r, 

«i     «2    Witiz— »2<^i      Wi     W2     tiidj      ^      br       ^  „  J 

-r—T==—l — J — '    -T^l-=1 — '    P==  — ,  etc.,  are  called 

formulas;  because  they  express  laws  of  number  in  brief  form, 
that  is,  they  formulate  laws  of  number. 


1 88 


Pirst-  Year  Mathematics 


Graphing  Percentage  and  Interest 
135.  The  laws  of  percentage  may  be  more  clearly  shown 
graphically. 

Percentage 

1.  What  is  2%  of  $100 ?    Of  $200  ?    Of  $300 ? 

2.  Let  a  vertical  side  of  a  small  square  (Fig.  198),  denote 
$2,  and  mark  upward  from  the  horizontal  O  X,  on  the  verticals 


^ 


0       jM*  £tif    |M«  Aim* 

Fig. 


through  $100,  $200,  $300,  $400,  etc.,  the  percentages  at  2% 
of  $100,  $200,  $300,  etc.,  as  shown.  Draw  a  line,  as  O  P, 
through  the  tops  of  the  lines.  What  kind  of  line  does  O  P 
seem  to  be  ? 

3.  Starting  again  on  the  $100- vertical,  mark  off  similarly 
distances  above  O  X  to  show  4%  of  $100,  of  $200,  of  $300,  of 
$400,  etc.,  to  $500,  and  connect  the  upper  ends  with  a  con- 
tinuous line,  as  O  M. 

4.  Draw  a  similar  line  for  the  percentages  at  3%  of  the 
same  amounts. 

5.  Draw  a  similar  percentage-line  for  1%  of  the  same 
amounts. 

6.  Through  what  point  do  all  these  percentage-lines  pass  ? 
What  does  this  mean  ? 


Practice  in  Algebraic  Language 


189 


7.  How  does  the  percentage-line  at  3%  lie  with  respect  to 
the  lines  at  2%  and  at  4%  ? 

8.  Show  that  the  line  O  P  might  be  represented  by 

percentage  =  .02  X  base,  or  more  briefly,  by 
p=  .02X&,  or  p=-^b. 

9.  Write  similar  equations  for  the  4%-line;  the  3%-line; 
the  1%-line. 

10.  Write  similar  equations  for  a  5%-line;  a  10%-line; 
a  6%-line;  and  an  r%-line. 

11.  In  Fig.  lyS  what  does  the  h*ne,  AB,  drawn  midway 
between  the  $300- vertical  and  the  $400- vertical  from  O  X  up 
to  O  P,  represent  ?    What  would  A  C  represent  ?    E  F  ?    EG? 

12.  How  would  you  read  from  the  figure  the  percentage  at 
4%  for  $1 50  ?    For  $250  ?    For  $450  ? 

13.  Draw  a  number  of  equally  spaced  verticals  upward 
from  the  horizontal  and  mark 
the  lines  0%,  1%,  2%,  etc.,  as 
shown.  Taking  $500  as  a  base, 
measure  upward  from  O  X  on 
the  1%-vertical,  1%  of  $500,  or 
$5,  letting  a  side  of  a  small 
square  division  denote  $5.  This 
gives  the  point  a.  On  the  2  %-line 
measure  and  mark  oflf  2%  of  $500;  on  the  3%-line,  3%  of 
$500,  and  so  on.     Connect  a,  b,  c,  d,  etc.,  by  a  line. 

14.  How  could  one  read  from  the  drawing  2^%,  2 J%,  4f%, 
....  of  $500  ? 

15.  Taking  a  base  of  $250  construct  a  $25o-percentage  line. 

16.  Construct  a  $1  co-percentage  line. 

17.  Show  that  the  $5oo-percentage  line  might  be  expressed 
as  an  equation,  thus: 

Percentage = $500  X  rate, 

$Soor 
or  p=^ — , 
100 

or  /»=$5r. 


»»  j«  ^1  $'%  ix  t%  1%  f%ji%X 

Fig.  199 


190 


First- Year  Mathematics 


18.  Write  a  similar  equation  for  a  $ioo-percentage  line; 
a  $4oo-percentage  line. 

19.  All  three  lines  go  through  the  o-point.    What  does 
this  signify  ? 

136.  The  laws  of  simple  interest  may  be  shown  and  interest- 
problems  may  be  solved  graphically. 

1.  Find  the  interest  on  $100  at  3%  for  i  year;    2  years; 
3  years;  4  years. 

2.  Call  a  side  of  a  small  square  $3,  mark  a  point,  as  a, 
to  show  $3  on  the  i-year  vertical.    Mark  off  $6  on  the  2 -year 

vertical,  as  at  b;  $9  on  the  3-year 
vertical,  etc.  Through  the  points 
a,  b,  and  c,  draw  a  straight  line  and 
extend  it.     This  is  the  3%-line. 

3.  Tell  from  the  drawing  the 
interest  on  $100  for  i|  years  at 
3%;  for  2 J  years  at  3%;  for  3^ 
years  at  3%. 

4.  Calculate  the  interest  on  $100 
at  5%  for  I  year;  2  years;  3  years; 
4  years;  etc.,  and  draw  the  5%-line. 

5.  From  the  5%-line  give  the  interest  at  5%  on  $100  for 
li  years;  for  2^  years;  for  3^^  years. 

6.  Calculate  three  points  and  draw  the  8%-line. 

7.  All  these  lines  go  through  the  zero-point,  o.     What  does 
this  mean  ? 

8.  Explain  how  the  lines  show  that  the  interest  at  a  given 
rate  on  $100  increases  proportionally  to  the  time. 

Such  work  as  that  above  is  called  plotting,  or  graphing. 
Percentage-lines  and  interest-lines  are  called  graphs. 

137.  Graphing  Statistics  Expressed  as  Percentage. 

I.  Taking  the  average  price  of  food  in  the  United  States 
from  1899  to  1900  as  100,  the  relative  prices  from  month  to 


/ 

/ 

/ 

/ 

/ 

'^^ 

\A 

> 

".4 

/ 

t    a 

3             ( 

r«< 

r  M 

n». 

Fig.  200 


Practice  in  Algebraic  Language 


191 


month  for  1905-6  and  1907  are  given  below.  Graph  the 
values  for  1905-6-7  on  a  sheet  of  rectangle  paper  as  shown 
for  1905.     Plot  the  tenths  by  estimate. 


Month 

190S 

Year 

igoe 

1907 

January 

115. 6 

117. 0 

121 .6 

February 

115. 8 

116. 2 

121 .1 

March 

113. 8 

115. 0 

119. 6 

April 

112. 0 

II3-9 

118. 3 

May 

no. 7 

113. 0 

117. 6 

June 

109.8 

113. 0 

117. 8 

July 

109.7 

II3-3 

118. 4 

August 

no. I 

113. 8 

119-3 

September 

no. 8 

115-2 

121 .1 

October 

112. 1 

117. 1 

123.4 

November 

113. 8 

119-4 

124.2 

December 

115-2 

121 .2 

125.0 

Hi 

m 
m 
m 
m 
m 
1/2 
w 
m 

M 

m 
m 

KM 

J 

( 

r 

/ 

r" 

s. 

/ 

/ 

V 

/ 

/ 

\, 

J 

/ 

\ 

t.  _> 

/ 

I 

V-*^ 

\ 

j 

-A 

J 

\ 

I 

\ 

/ 

% 

/ 

\, 

/ 

Si ,' 

■^ 

\ 

/ 

V 

y 

\^ 

>.<^ 

>fc- 

■^ 

190S 

rt        Ja^       Af,r        JUly        Oct        Ja^         A 

me 

fr.      July       Oct.    Dt             I 
/W7                             J 

Fig.  201 


192 


First- Year  Mathematics 


2.  When  did  the  price  of  food  reach  the  highest  point? 
The  lowest  ? 

3.  In  what  month  does  the  price  of  food  seem  to  reach  a 
maximum  ?    A  minimum  ? 

4.  What  is  the  general  trend  of  prices,  a  rise  or  a  fall  ? 

5.  Using  1 00  for  the  average  price  of  food  for  1890  to 
1899,  the  relative  prices  for  the  years  1890  to  1907  are  given 
here.     Graph  the  values  on  rectangle  paper. 


Year 

Per  cent. 

Year 

Per  cent. 

Year 

Per  cent. 

1890 

102.4 

1896 

95-5 

1902 

I10.9 

1891 

103.7 

1897 

96.4 

1903 

no. 2 

1892 

102.0 

1898 

99.8 

1904 

III. 7 

1893 

104.3 

1899 

99.6 

1905 

112. 4 

1894 

99-7 

1900 

lOI  .0 

1906 

II5-7 

189s 

97.8 

1 901 

105. 1 

1907 

120.7 

m 

y 

/ 

. 

/ 

' 

M 

m 

. 

'      J 

^ 

m 
m 
m 

Si 
X 
a* 

■ 

1 

^ 

1 

\ 

1 

_ 

' 

/ 

■, 

^ 

^ 

; 

^ 

\ 

' 

luur.. 

K  /»  fflii  ilii  /«« ;ws  im  im  im  n»  »oo  1901  i^n  i»M  r}<x  ihu  0«i  a 

n 

Fig.  202 

6.  When  did  the  price  of  food  reach  a  maximum  ? 

7.  When  did  the  price  of  food  reach  a  minimum  and  then 
rise? 


Practice  in  Algebraic  Langtiage 


193 


138.  Graphing    "Sun    Fast"  and  "Sun  Slow"  Data  of  the 
Almanac. 

Almanacs  say  "sun  fast"  and  "sun  slow"  according  as 
the  sun,  in  its  daily  motion,  reaches  the  meridian  ahead  of  or 
behind  the  average  time  for  the  year.  There  are  two  causes 
of  this  difference : 

I.  The  speed  of  the  earth  is  different  at  different  places  in 
its  orbit,  and  this  makes  the  sun  appear  to  move  at  different 
rates; 

II.  The  apparent  path  of  the  sun  is  along  the  ecliptic, 
which  is  inclined  23^°  to  the  celestial  equator  along  which 
time  is  measured. 

The  amounts  due  to  the  first  cause  are  given  for  15-day 
periods  in  the  columns  headed  "I."  When  the  sun  is  ahead 
of  its  average  time,  the  number  is  + ,  when  behind  — . 

I.  Graph  these  numbers  to  a  convenient  scale  on  cross- 
lined  paper,  and  draw  through  the  points  graphed  a  smooth 
free-hand  curve. 


Date 

Jan.      I 
16 

Mint 

0 

+  2 

[ 

ites 
00 

50 

II 

Minutes 

+  3-75 
+  7-50 

Date 

July     I 

16 

MiB 

0 

—  2 

I 

utes 
00 

50 

I 

Min 

+  3 
+  7 

I 

utes 

75 
50 

Feb.      I 
16 

+3 
+  5 

75 
50 

+8.50 
+8.50 

Aug.     I 

16 

-5 
-6 

00 
50 

+9 
+  9 

SO 
50 

Mar.     I 
16 

+6 

+7 

50 
50 

-1-7.00 
+  1.25 

Sept.    I 
16 

-7 
-8 

50 

50 

+  7 
+  2 

50 
50 

April     I 
16 

-1-8 
+8 

25 
25 

—  5.00 
-8.50 

Oct.     I 
16 

-8 
-8 

55 
50 

-5 
-8 

00 

75 

May     I 
16 

+  7 
+6 

SO 
25 

—  1 0 . 00 
—  9.00 

Nov.    I 
16 

-7 
-6 

50 
50 

—  10 
-8 

00 

75 

June     I 

+  5 

00 

—  7 .00 

Dec.     I 

-5 

00 

-6 

25 

16 

+  2 

25 

—  2.00 

16 

—  2 

SO 

—  2 

00 

2.  The  amounts  due  to  the  second  cause  are  in  the  columns 
headed  "II."     Graph  these  values  on  the  same  axes  and  to 


194 


First-  Year  Mathematics 


the  same  scale  as  was  used  for  the  data  of  columns  "I,"  and 
draw  through  the  points  a  smooth  free-hand  curve. 


9Cm 

■" 

— 

— 

— 

— 

— 

■~" 

p- 

" 

n 

"■ 

' 

■" 

-- 

... 

~' 

"■ 

1 

/ 

/ 

"^ 

\ 

/ 

) 

1 
/ 

1 

\ 

-- 

^ 

^ 

\ 

1 

/ 

'; 

/ 

/ 

/ 

\ 

\ 

\ 

i 

/ 

"1 

N 

\ 

.     ■ 

-5m 

/ 

F 

H 

\ 

a' 

V 

M 

y 

^ 

\ 

\ 

A 

\ 

V 

0 

V 

D 

I 

) 

:_ 

,« 

',' 

'' 

\ 

s 

s 

\ 

J< 

r^ 

/ 

y 

/J 

! 
/ 

\ 

^ 

I 

/ 

\ 

k^ 

/ 

/ 

Fig.  203 

3.  Add,  algebraically,  the  corresponding  verticals  of  the 
two  curves  and  draw  a  curve,  through  the  points  located  by 
the  sums  of  the  verticals,  showing  the  combined  effect  of  the 
two  causes  operating  together. 


The  Evaluation  of  Expressions* 

139.  The  formulas  used  in  the  following  problems  express 
laws  of  motion  and  mensuration. 

*  The  general  equations  of  this  section  might  well  be  pictured  by 
means  of  diagrams  on  cross-lined  paper  or  cross-ruled  blackboard,  or 
by  measuring  drawings  made  or.<  unruled  paper.  Many  of  the  laws 
may  be  graphed  with  profit. 


Practice  in  Algebraic  Language  195 

1 .  The  distance  traversed  by  a  moving  body  is  equal  to  the 
rate  multiplied  by  the  time;  that  is,  D=rt. 

Find  D,  if  (i)  r  =  30  ft.  per  second,  and  f  =  5  seconds; 
(2)  r  =  $  mi.  per  hour  and  ^  =  17  hours. 

2.  The  area  of  a  rectangle  is  equal  to  the  product  of  the 
base  and  the  altitude;  that  is,  A  =ba. 

Find  A,  if  (i)  6  =  13  ft.,  and  a  =  24  feet; 

(2)  &  =  io.2  in.,  and  a =3. 5  inches. 

3.  The  area  of  a  triangle  is  equal  to  ^  the  product  of  the 
base  by  the  altitude;  that  is,  A  =^ba. 

Find  A,  if  (i)  6  =  12  ft.,  and  a  =  i6  feet; 

(2)  6=8.2  rd.,  and  a  =  7. 78  rods. 

4.  The  area  of  a  parallelogram  is  equal  to  the  product  of 
the  base  by  the  altitude;  that  is,  A  =ba. 

Find  A,  if  (i)  6  =  28,  and  a  =  ig; 

(2)  &  =  i6. 3,  and  0  =  14.6. 

5.  Find  the  numerical  value  of  each  of  the  following  ex- 
pressions, for  0  =  5,  6  =  3,  c  =  io,  w=4,  «  =  i; 

(i)  <,cm'  ,      c+2m 

(2)  3a3c  c—2m 

(3)  i36^cw2  (7)  a'+b'+c'+m'+n' 

(4)  «s  +  i  (8)  2ab-{-4bc+scm' 

(5)  ^abc+T,m''n3  (9)  a^—b^-j-c^ 

(10)  ^abcmn . 

6.  Two  weights,  Wi  and  W2,  will  balance  on  a  beam  that 
lies  across  a  stick  when  the  distances,  dj  and  dj,  of  the  weights 
from  the  stick  are  in  the  inverse  ratio  of  their  weights;  i.  e., 

when  -J-  =— ^  . 

Find  <ii,  if    (i)  (^2  =  18  ft.,    ^2=60  lb.,     1^1=50  lb. 
(2)  ^2  =  27  in.,    W2  =  36  1b.,    ^1=24  lb. 

Find  (^2,  if    (i)  £^1=40  in.,    ^2  =  16  lb.,    Wi  =  i8  1b. 
(2)  (^,=25  in.,    ^2  =  3-8  lb..  Wi  =  2.85  lb. 


196  First-Year  Mathematics 

Find  Wx,  if  (0  ^1  =  2.5  in.,  d,  =  T.sit;  Wa  =  io.5lb. 
(2)  di=6.6h.,  ^3=9. 9  ft.,  ^2  =  17  lb. 

Find  w„  if  (i)  rf,=3-5ft.,  d^=S.sit.,  w,  =  3olb. 

7.  The  weight,  w,  of  any  mass  is  equal  to  the  volume,  t 
(number  of  cubic  inches  it  holds),  multiplied  by  the  density 
(the  weight  of  one  cu.  in.);  i.  e.,  w=vd. 

Find  w,  if  z;=64  cu.  in.,  and  d  =  i6.2  lb. 
Find  v,  if  ^=648  lb.,  andrf  =  i6.2lb. 
Find  rf,  if  xf =800,  and  ^'  =  I6o. 

8.  The  weight,  w,  that  a  force,  p,  pulling  up  a  smooth 
slope  h  feet  high  and  /  feet  long  will  just  move,  is  given  by 

/ 

Find  h,  if  •w=j20  lb.,  /  =  2o  rd.,  p=6o  lb. 

9.  A  stone,  falling  from  rest,  goes  in  a  given  time  16 
ft.  multiplied  by  the  square  of  the  number  of  seconds  it  has 
fallen;  i.  e.,  5  =  16/*. 

Find  5,  if  (i)  t=/\.  seconds 

(2)  /  =  ii  .5  seconds. 

Find  t.\i  (i)  5=64  feet 

(2)  5  =  1,600  feet. 

10.  A  stone,  thrown  downward,  goes  in  a  given  time  16 
ft.  multiplied  by  the  square  of  the  number  of  seconds  it  has 
fallen,  plus  the  product  of  the  velocity  with  which  it  is  thrown 
and  the  number  of  seconds  fallen:    i.e.,  s  =  i6t^+vt. 

Find  5,  if    (i)  /=i2  seconds,  and  v  =  ^  ft.  per  second 
(2)  t=  8  seconds,  and  v=y  ft.  per  second. 

Find  V,  if   (i)  /  =  5      seconds,  and  5  =  500  feet. 
(2)  /=!  .5  seconds,  and  s=  60  feet. 

1 1 .  The  time,  /,  taken  for  a  pendulum  to  make  a  single  vibra- 
tion equals  ttaI -  ,  where  /  is  the  length  of  the  pendulum  in 
feet,  ^  is  32,  and  tt  is  V^. 


Practice  in  Algebraic  Language  197 

Findf,  if  (i)/=8ft. 

(2)  /  =  f  11- feet. 

Find/,  if  (i)  t  =  i  second 
(2)  t=4  seconds. 

12.  Tiie  area,  ^4,  of  a  triangle  is  equal  to 

Vs{s-a){s-b){s-c)  , 
where  a,  b,  c  represent   the  lengths  of  the  sides  and  5= ^(a 
+6+C). 

Find  ^,  if  (i)  0  =  3,   b=  4,  c=  5 
(2)  a  =  s,   b  =  i2,  c  =  i3. 

13.  h= — .  Find  A,  if  (i)  7;=ii,  ^=32.2 

(2)  v=  2,  ^=32.2. 

14.  R=~:^,'    Find  iJ,  if  (i)  r  =  ii.s,   ^'=6.5 

(2)  r=i3,       r'  =  i5. 

15.  £  =  -^^  .     Find  E,  if  (i)  M  =  i2,   F  =  5 

(2)  M  =  ii,   V=g 
Find  M,  if  (i)    E=  8,   V=4 
(2)    £  =  50,   F  =  5. 

16.  V=^h{B-\-b  +  l/Bb). 

Find  F,  if  (i)  h=g,  B  =  i6,   6=4 
(2)  ;j=6,   5  =  16,   6=9. 

140.  The  formulas  used  here  express  properties  of  the 
circle  and  the  sphere. 

1.  The  circumference  of  a  circle  is  equal  to  ^  of  the 
diameter;  i.  e.,  C=7r  ■  d,  where  ir  =  '^. 

Find  C,  if  (i)  d  =  2\  ft.  Find  d,  if  (i)  C=88  ft. 

(2)  rf=  7  ft.  (2)  C=66ft. 

(3)^=ift.  (3)C  =  i6ft. 

2.  If  r  stands  for  the  radius  of  a  circle  and  C  for  its  cir- 
cumference, from  problem  i,  show  that  C  =  2irr. 


198 


First-  Year  Mathematics 


Find  C,  if  (i)  r=  3  ft.  Find  r,  if  (i)  0  =  132  in. 

(2)  r=  6  ft.  (2)  C=  66  ft. 

(3)  f=i8ft.  (3)  C  =  iiord. 

3.  The  area  of  a  circle  equals  ^  times  the  square  of  the 
radius;  i.  e.,  A=irr',  where  ■ir=^%^-. 

Find  A,  if  (i)  f=  3  ft.  Find  r,  if  (i)  ^=154  sq.  ft. 

(2)  r=  7  ft.  (2)  A=22o  sq.  ft. 

(3)  r  =  2ift.  (3)  ^=86.4  sq.ft. 

4.  The  volume  of  a  sphere  equals  fir  times  the  cube  of 
the  radius;  i.  e.,  V^^vr^,  where  ir=^. 

Find  V,  if  (i)  r  =1  ft.  Find  r,  if   (i)  7=42^  cu.  in. 

(2)  r=iift.  (2)  F  =  33H-cu.ft. 

(3)  r=  2  ft.  (3)  V  =  s^^T  cu.  rd. 

5.  The  circumference,  C,  the  area,  i4,  of  a  section  through 
the  center,  the  area,  5,  of  the  surface,  and  the  volume,  V,  of  a 
sphere,  are  connected  with  the  radius,  r,  of  the  sphere  by  the 
formulas: 

C  =  2irr;  A=Trr';  S=4-Tr';  and  F=|7rr3. 

Find  C,  A,  S,  and  V  for  a  sphere  whose  diameter  is  i  yard. 

Find  A,  S,  V,  and  r,  for  C  =  iSir. 

Find  C,  S,  V,  and  r,  for  ^  =367r. 

Find  C,  A,V,  and  r,  for  5  =  ioc»r. 

Find  C,  A,  S,  and  r,  for  F  =  367r. 

The  Right  Triangle 
141.  The  sides  of  a  right  triangle  are  connected  by  a 
simple  law. 

I.  Fig.  204  shows  a  right  tri- 
angle whose  sides  are  3,  4,  and  5. 
How  does  the  sum  of  the  squares  on 
the  sides  including  the  right  angle 
compare  with  the  square  on  the  side 
opposite  the  right  angle  (the  hypot- 
enuse) ? 
Fig.  204  2.  How   does   the   sum    of  the 


Practice  in  Algebraic  Language 


199 


squares  on  the  sides  of  the  right  triangle  of   Fig.  205  com- 
pare with  the  square  on  the  hypotenuse  ? 


V 

X 

X 

X 

f7\ 

X 

3>< 

7X5 
/6\ 

\j/ 

X 

V 

/3\ 

X 

X 

X 

X 

X 

X 

Fig.  205 


Fig.  206 


3.  Answer  a  similar  question  for  the  right  triangle  of 
Fig.  206. 

4.  Express  this  relation  for  the  triangle  of  Fig.  206  by 
means  of  an  equation. 

142.  Theorem:  In  any  right  triangle  the  sum  of  the 
squares  on  the  sides  including  the  right  angle  is  equal  to  the 
square  on  the  hypotenuse. 

1 .  The  two  short  sides  of  a  right  triangle  are  6  and  8.  What 
is  the  length  of  the  hypotenuse  ? 

2.  A  wall  is  16  ft.  long  and  12  ft.  high.  How  long  is  a 
string  stretched  from  a  lower  comer  to  the  opposite  upper 
comer  ? 

3.  The  hypotenuse  of  a  right  triangle  is  30  ft.,  and  one  of 
the  other  sides  is  18  ft.  long.  How  long  is  the  third 
side? 

4.  The  sides  of  a  right  triangle  are  as  3  is  to  4  and  the 
hypotenuse  is  35  ft.  long.    How  long  are  the  sides? 

5.  A  ladder  20  ft.  long  just  reaches  a  window  16  ft.  above 
the  ground.  How  far  is  the  bottom  of  the  ladder  from  the 
foot  of  the  wall  if  the  ground  is  level  ? 


200  First-Year  MatJtematics 

6.  The  diagonal  of  a  square  is  12.  What  is  the  side? 
The  perimeter  ? 

The  square  root  of  a  number,  as  80,  or  a,  or  x-^y,  that  is  not  a 
perfect  square,  is  indicated  thus:    P  80,  or  V  a,  or  V  x-\-y. 

7.  The  diagonal  of  a  square  is  a.     Find  the  perimeter. 

8.  Find  the  hypotenuse,  h,  of  a  right  triangle  whose  area  is 
54  sq.  ft.  and  whose  base  is  12  feet. 

9.  Find  the  hypotenuse,  h,  of  a  right  triangle  whose  area 
is  r  sq.  rd.  and  whose  base  b  h  rods. 

10.  Find  the  perimeter  of  a  right  triangle  whose  area  is 
216  sq.  rd.  and  whose  base  is  48  rods. 

11.  Find  the  perimeter  of  a  right  triangle  whose  area  is 
5  sq.  rd.  and  whose  base  is  h  rods. 

12.  A  tree,  standing  on  level  ground,  was  broken  45  ft. 
from  the  top  and  the  top  struck  the  ground  27  ft.  from  the 
stump,  the  broken  end  remaining  on  the  stump.  How  tall 
was  the  tree  before  breaking  ? 

13.  Under  conditions  of  problem  12,  suppose  the  top  piece 
/  ft.  long  and  that  the  top  struck  the  ground  /  ft.  from  the 
stump.    How  tall  was  the  tree  before  it  broke  ? 

The  Circle 

143.  A  circle  is  a  curve,  all  of  whose  points  are  at  the  same  dis- 
tance from  a  fixed  point,  as  C.     The  length  of  the  curve  is  the 

circumference.   The  fixed  point 
C,  is  the  center  of  the  circle. 
A  chord  is  a  line  connect- 
ing two  points  of  a  circle. 

The  word,  line,  here  means 
straight  line. 

A  radius  is  a  line  connect- 
ing the  center  with  a  point  on 
the  circle. 

A    diameter    is    a    chord 
Fig.  207  through  the  center. 


Practice  in  Algebraic  Language  201 

Prove  that  a  diameter  is  twice  as  long  as  a  radius. 

A  secant  is  a  line  cutting  the  circle  in  two  points. 

A  tangent  is  a  line  that  touches,  but  does  not  cut,  the  circle, 
however  far  produced.  The  point,  as  P,  where  a  tangent 
touches  the  circle  is  called  the  point  of  contact,  or  the  contact- 
point. 

A  radius,  drawn  to  the  contact-point  of  a  tangent,  is  per- 
pendicular to  the  tangent. 

1.  Find  the  angle  between  the  tangent,  PT,  and  the  chord 
P  C,  if  the  angles   may   be   designated   as 
shown  in  Fig.  208. 

2.  Draw  a  square,  with  its  sides  tangent 
to  a  circle,  and  show  that  the  area  of  a  circle 
is  less  than  4/"^. 

A  square  whose  sides  are  tangent  to  a 
circle  is  a  circumscribed  square.  ^^^-  ^°^ 

A  square  whose  sides  are  chords  of  a  circle  is  an  inscribed 
square. 

3.  Draw  a  square  inside  a  circle,  with  the  comers  on  the 
curve,  and  show  that  the  area  of  a  circle  is  greater  than 
2r^. 

The  expression,  2r^<.A<.^r'  is  read:  "Two  r-square  is 
less  than  A  is  less  than  four  r-square."  It  means  that  A  is 
between  2r^  and  4^^  in  magnitude. 

The  symbol,  < ,  means  is  less  than. 

The  symbol,  > ,  means  is  greater  than. 

The  length  of  the  curve  of  a  circle  is  about  3^  times  the 
diameter.  More  accurately,  the  ratio  is  3. 141 59.  For  con- 
venience, the  symbol  ir  (called  pi)  is  often  used  to  denote  the 
correct  value  of  the  ratio  of  the  circle  to  its  diameter. 

The  Triangle 

144.  To  find  the  distance  straight  through  a  hill  from 
A  to  B,  a  point,  C,  was  chosen  on  level  ground,  so  that  the 


203 


Fir  St- Year  Mathematics 


distances,  A  C  and  B  C,  could  be  measured.     C  D,  in  the 
prolongation  of  B  C,  was  made  equal  to  B  C,  and  C  E,  in  the 

prolongation  of  AC,  was 
made  equal  to  A  C.  The 
points,  D  and  E,  were  marked 
with  stakes,  and  DE  was 
measured  and  found  to  be 
68.5  rods. 

1 .  How  long  is  A  B  ? 

2.  What  is  the  relation  of 
angles  x  and  y  ? 

3.  Recalling  how  D  and  E 
were  located,  what  parts  (angles  and  sides)  of  triangle  ABC 
are  known  to  be  equal  to  certain  parts  of  triangle  C  D  E  ? 

4.  What  relation  as  to  form  and  size  do  the  whole  triangles 
ABC  and  C  D  E  seem  to  have  ? 

145.  Theorem:  If  two  sides  and  the  included  angle  of  one 
triangle  are  equals  each  to  each,  to  two  sides  and  the  included 
angle  of  another  triangle,  the  triangles  are  equal  in  all  respects. 

Draw  a  triangle,  as  A  B  C,  and  make  another  triangle  as 
A'B'C,  having  a'=a,  c'=c,  and  angle  5= angle  B'. 

Trace  triangle  ABC  through  a  piece  of  thin  paper,  or  on 
tracing  paper,  and  fit 
the  trace  over  triangle 
K'B'C  by  placing  the 
trace  of  a  along  a',  of 
B  on  B',  and  of  c  on  c'. 
What  seems  to  be  true 
of  the  relations  of  h  and  h'  ? 
and  AC  7 

It  is  not  necessary  always  actually  to  do  all  this  fitting  and 
measuring.  It  is  quicker  and  better  to  reason  out  the  equality 
of  the  triangles,  thus: 

Imagine  triangle  ABC  carried  along  and  placed  over  tri- 


Of  AA  and  Z.A'}    Of  AC 


Practice  in  Algebraic  Language 


203 


angle  A'B'C,  so  that  point  B  falls  on  point  B',  and  c-  lies 
along  a'. 

As  angle  5= angle  B',  c  falls  along  c'. 
As  a=a',  point  C  must  fall  on  point  C 
As  c=c',  point  A  must  fall  on  point  A'. 
h  and  b'  must  then  be  one  and  the  same  straight  line  (i.  e., 
they  must  coincide),  because  only  one  straight  line  can  be  drawn 
connecting  two  points. 

Consequently,  the  two  triangles,  ABC  and  A.'WC',  can  be 
made  to  fit,  or  coincide  throughout  and  hence  they  are  equal 
in  all  respects,  or 

If    a=a',  then,      b=b', 

c=c',  /LA  =  AA\ 

and  ZB=ZB\  and  ZC=ZC'. 

1.  If  a=a',  b=b',  and  Z.C=  AC,  compare  c  and  d,  Z.A 
and  Z.A\  AB  and  AB' . 

2.  If  b=b',  c=^c\  and  /.A  =  Z.A',  compare  a  and  a', 
AB  and  AB',  AC  and  AC 

3.  With  the  triangles  of  Fig.  211  prove  that 

(i)  if  a=a\  c=c',  and  y=y';   then  b=b',  x=x',  and  z=z^ 

(2)  if  c=c',  b=b',  and  x=x';   then  a=a',  z=z',  and  y=y 

(3)  if  a=^a',  b=y,  and  z=z';   then  c=c',  x=x',  and  y=y'. 
A  triangle  that  has  two  equal  sides  is  an  isosceles  triangle. 


4.  Draw  the  bisector  of  the  angle  A  of  the  isosceles  tri- 
angle, Fig.  212  (c=6),  and  prove  that  angle  jB=  angle  C. 

5.  Prove  that  the  angle-bisector  of  angle  A  bisects  a,  at  D. 


204 


First- Year  Matliematics 


6.  Prove  that  the  angle-bisector  of  A  is  perpendicular  to 
side  a. 

7.  Prove  that  if  the  sides  including  the  right  angle  of  a 
right  triangle  are  equal,  each  to  each,  to  the  sides  including 
the  right  angle  of  another  right  triangle,  the  triangles  are  equal 
in  all  respects. 


Fig.  213  Fig.  214 

8.  The  span  of  a  roof  is  48  feet,  and  c  and  h  are  32-foot 
rafters.  What  is  the  rise  AD?  (A  D  is  the  angle-bisector 
of  A.) 

9.  The  span  of  a  roof  is  2s,  the  length  of  the  rafters,  r. 
Find  the  rise  h. 

146.  A  triangle  that  has  all  sides  equal  is  an  equilateral 
triangle. 

1.  Prove  that  an  equilateral  triangle  is  equiangular  (all 
angles  equal).     Apply  problem  4,  p.  203. 

2.  The  length  of  a  side  of  an  equilateral  triangle  is  a. 
Find  the  length  of  the  angle-bisectors,  shown  in  Fig.  215. 


3.  How  many  degrees  are  there  in  one  of  the  angles  of 
an  equilateral  triangle  ? 

Let  X  denote  the  value  of  one  of  the  angles,  then  3:x;  =  i8o°.     Why  ? 

4.  How  many  tiles  of  the  shape  of  equilateral  triangles 
will  just  cover  the  plane  if  placed  as  shown  in  Fig.  216  around 
a  point  as  O  ? 


Practice  in  Algebraic  Language 


205 


Fig.  217 


5.  Draw   6   equal   equilateral   triangles    in    positions   as 
shown  in  Fig.  217.     What  kind  of  figure 
is  formed  ? 

Compare  the  lengths  of  the  sides  of  the 
large  figure. 

Compare  the  angles  of  the  large  figure. 

Compare  the  distances  from  O  to  each 
of  the  comers  A,  B,  C,  etc. 

A  plane  figure  that  is  bounded  by  6  equal  sides  making 
equal  angles  is  a  regular  hexagon. 

6.  If  O  were  used  as  a  center  and  a  circle  were  drawn 
with  O  A  as  a  radius,  through  what  points  would  the  circle 
pass? 

7.  How  many  chords  equal  in  length  to  the  radius 
of  a  circle,  if  placed  end  to  end,  will  reach  around  the 
circle  ? 

8.  How  may  a  regular  hexagon  be  readily  constructed  by 
the  aid  of  a  circle  ? 

9.  Six  lights  are  placed  equal  distances  apart  around  a 
circle  of  radius  r.  Find  the  distances  from  any  one  light  to 
each  of  the  others. 

L  B  and  A  C  bisect  each  other  perpendicu- 
larly. The  distance  L  A,  Fig.  218,  is  the  radius 
of  the  regular  hexagon. 

The  distance  LN  perpendicular  to 
A  B,  Fig.  218,  is  the  apothem. 

10.  Find  the  area  of  an  equilateral 
triangle  whose  side  is  12. 

11.  Find  the  area  of  an  equilateral  tri 
angle  whose  side  is  a.     Fig.  219. 

12.  Find  the  area  of  a  regular  hexagon, 
one  of  whose  sides  is  10. 


Fig.  218 


2o6 


First-  Year  Mathematics 


Fig.  220 


13.  Find  the  area  of  a  regular  hexagon  (Fig.   220),  one 
of  whose  sides  is  5. 

14.  A  wheel  of  85  cogs  works  into  a 
wheel  of  27  cogs.  In  how  many  revolu- 
tions, X,  of  the  larger  does  the  smaller 
gain  g  revolutions  ? 

15.  A  wheel  of  a  cogs  works  into  a 
wheel  of  h  {a>b)  cogs.  In  how  many 
revolutions,   x,  of  the   larger,    does   the 

smaller  gain  g  revolutions? 

16.  Draw  two  concentric  circles  of  radii  6  and  10.     Find 

the  length,  /,  of  the  chord  of  the  outer  circle  that  is  tangent  to 

the  inner  circle. 

A  radius,  drawn  to  the  contact-point  of  a  tangent,  is  perpendicular 
to  the  tangent.     See  p.  201. 

17.  Draw  two  concentric  circles  of  radii  r  and  R  {r<.R). 
Find  the  length,  /,  of  the  chord  of  the  outer  circle,  that  is 
tangent  to  the  inner  circle. 


Fig.  221 


Fig.  222 


If  the  area  of  a  circle  be  divided  by  drawing  radii,  as  in 
Fig.  222,  into  small  triangles,  having  curved  bases,  the  sum  of 
the  areas  of  all  these  triangles  equals  the  area  of  the  circle. 

18.  Denoting  the  radius  of  the  circle  by  r,  and  the  small 
bases  by  6„  b^,  63,  b^,  etc.,  express  the  sum,  S,  of  the  areas 
of  the  triangles. 

19.  To  what  curved  line  is  the  sum  bj-\-b2+b^-\- 

equal  ? 

The  series  of  small  dots, ,  means  "and  so   on."      It   is 

called  a  symbol  oj  conlinuathn. 


Practice  in  Algebraic  language  207 

20.  Show  that  the  area,  A,  of  the  circle  is  given  by 

^=5=-.(6x+6.+&3  + )=^.C='^=r.-, 

2  222 

C  denoting  the  length  of  the  circle. 

21.  State  in  words  the  meaning  of 

1.  A=S  3.  A=--C 

^  2 

2.  A=r-—  4.  A= . 

2  2 


Summary 

Algebraic  expressions  are  only  translations  of  verbal  state- 
ments of  number  relations  into  compact  symbolic  forms. 

Algebraic  expressions  need  often  to  be  translated  into 
verbal  forms  and  verbal  expressions  of  number  relations  into 
algebraic  or  symbolic  forms. 

The  fundamental  operations  of  arithmetic  may  be  sym- 
bolized thus: 

5=ai-|-a2+a3  .  .  .  -ho»  for  addition  of  «  addends 
d  =  m—s  for  subtraction 

P=M  •  m  for  multiplication 

Q=D-i-d  for  division. 

In  division,  when  there  is  no  remainder,  D=Qd. 
In  division,  when  there  is  a  remainder,    D  =  Qd-\-r. 

Adding  two  fractions  is  indicated  by  ^+^=  -^—7 — 7-^-^  • 

Oi     di         d^-ds 

Subtracting  fractions  is  shown  by  -^  —j-  =  -^y — -p-^  . 

ft  ft  ft    ft 

Midtiplying  two  fractions  is  shown  by  'r-y^-r —~7~t^  • 

T-,.   .,.  r       •     1  1-1         .     Ui     n^    n^     d, 

Dtvtdtng  one  fraction  by  anotheris  shown  by  —  -;-—  =  v~  X — . 


2o8  First-  Year  Mathematics 

Percentage  is  shown  by  p= . 

.     prt 
Interest  is  shown  by  t=^—  . 

lOO 

Percentage  and  interest  problems  may  be  solved  graphically. 

Statistical  laws  may  be  shown  graphically. 

Problems  in  motion  and  mensuration  may  be  solved  by 
substituting  in  the  formulas  that  express  the  laws  of  motion 
and  mensuration. 

Problems  on  the  circle  and  the  sphere  may  be  solved  by 
substituting  in  the  formulas  that  express  the  properties  of  the 
circle  and  the  sphere. 

In  a  right  triangle  the  square  of  the  hypotenuse  is  equal 
to  the  sum  of  the  squares  on  the  other  sides. 

A  circle  is  a  plane  curve  all  of  whose  points  are  at  the  same 
distance  from  a  fixed  point,  called  the  center. 

The  circumference  is  the  length  of  the  circle. 

A  chord  is  a  straight  line  connecting  two  points  of  the 
circle. 

A  radius  (plural  ra'dl-I)  of  a  circle  is  a  straight  line  con- 
necting the  center  with  a  point  on  the  circle. 

A  diameter  is  a  chord  through  the  center. 

A  secant  is  a  straight  line  cutting  the  circle  in  two  points. 

A  tangent  is  a  line  that  touches  but  does  not  cut  the  circle, 
however  far  produced. 

A  circumscribed  square  is  a  square  whose  sides  are  tangent 
to  a  circle. 

An  inscribed  square  is  a  square  whose  sides  are  chords  of 
a  circle.  The  symbol  <  means  is  less  than,  and  >  means 
is  greater  than. 

If  two  sides  and  the  included  angle  of  one  triangle  are  eqtuU, 
each  to  each,  to  two  sides  and  the  included  angle  of  another 
triangle,  the  triangles  are  equal  in  all  respects. 


Practice  in  Algebraic  Language  209 

A  regular  hexagon  is  a  plane  figure  that  is  bounded  by 
6  equal  sides,  making  equal  angles. 

The  apothem  of  a  regular  hexagon  is  the  perpendicular 
distance  from  the  center  to  the  side  of  the  hexagon. 

The  side  of  a  regular  hexagon  is  equal  to  the  radius  of  the 
circumscribed  circle. 

The  radius  of  the  hexagon  is  the  radius  of  the  circum- 
scribed circle. 

The  symbol  of  continuation  is ,  meaning,  and  so  on 

according  to  the  same  law. 


CHAPTER  X 


THE  SIMPLE  EQUATION  IN  ONE  UNKNOWN 

147.  Algebraic  expressions  may  be  pictured  or  graphed. 
and  equations  may  be  regarded  as  having  been  obtained  by 
giving  to  the  expressions  the  particular  value  o. 

I.  A  man  walks  along  a  straight  road,  AB,  in  the  direc- 
tion of  the  arrows  (^)  at  the  rate  of  3  mi.  an  hour.  How 
far  from  the  house,  H,  is  he — 


4  hr.  before  reaching  it  ? 
3  hr.  before  reaching  it  ? 
2  hr.  before  reaching  it  ? 
I  hr.  before  reaching  it  ? 


1  hr.  after  reaching  it  ? 

2  hr.  after  reaching  it  ? 

3  hr.  after  reaching  it  ? 

4  hr.  after  reaching  it  ? 


;:::::::::::::::::::::::::::::::::::::::=E! 

d.     2 

'                    "                              ^^ 

_              _                             ,-      _ 

-    _      -           ^^'              : 

_: : ^?_  _     .it 

_    ■                                       ^'^ 

I                        -              ^^ 

^■^ 

,n ^-  _i_ ,<.1. Z,<  *«_i.i.a.£.- , , 

'*3 ____-,j ^_a.___ g ^ 

-^ 

p  -            I              ,^'        -            I 

/  -              _                -^ 

_-,■'_ 

I  '^"^ _:i  I : : 

J ^^ 

iS  1:4:      ,^                                it 

\  -        ^^^                                 _ 

^Z I 

__^iL_ I 1 

i    .£r.=zz=======z=.r.==zz=z...==z.z.=.=z.=== 

Fig.  223 

2.  Show  the  times  to  a  scale  (i  large  space  =  1  hour)  on  a 
horizontal  line,  and  at  each  time  draw,  or  measure,  a  vertical 
downward  to  represent  to  scale  (i  small  space  =  1  mile)  his 
distances  below  H,  and  draw  verticals  upward  to  represent  his 
distances  above  H. 


The  Simple  Equation  in  One  Unknown 


211 


The  row  of  points  at  the  ends  of  the  verticals  pictures  the 
iuccessive  positions  of  the  man  with  respect  to  the  house,  H. 

3.  Show  on  the  graph  how  far  the  man  was  from  the  house 
at/=+Jhr.     At^=-ihr.    At /=o. 

4.  Show  on  the  graph  when  the  man  was  7  miles  below 
H;  4  miles  above  H, 

148.  By  putting  in  dots  more  thickly  more  distances  from 
H  would  be  represented  and  the  line  through  all  the  points 
represents  all  distances  of  the  man  from  H  between  —4  hr.  and 
+4hr. 

Since  the  distances  from  H  for  all  the  times  from  /  hours 
before  to  t  hours  after  arriving  at  H  are  also  represented  by 


,; 

:::::::::: :;'^  -: 

.--     -      --    -  ---  :  -Z^.i    :: 

. 

:":    ::    :::  ;  ::    'y/.m  ::  ::    :: 

:  -^                _  ..  -. 

,i 

:[::-j"~"-»"":^!j — z----r---fr--i\ 

^                        . 

K . 

: ,,:'^__:        _              __  : 

±;^    —           . 

.._E !!____::::_:::::::::::::_:::---. 

\itzzz. : ::::::::::: 

Fig.  224 


3^,  the  line  drawn  through  the  row  of  points  also  pictures  the 
expression  3/.  The  Hne  through  the  points  is  called  the 
graph  of  3/. 

If  the  distances  of  the  traveler  were  expressed  from  a 
point,  P,  3  miles  below  H,  the  row  of  points  and  the  line 
through  them  would  be  as  in  Fig.  224.  The  algebraic 
expression  of  these  same  distances  is  3/4-3.  The  line  through 
the  dots  of  Fig.  224  is  the  graph  of  3^+3. 


212  First- Year  MatJietnatics 

149.  Of  course,  any  other  letter,  as  x,  might  be  used  to 
stand  for  the  times.  If  x  had  been  used  instead  of  /,  the  line 
of  dots  of  Fig.  224  would  have  been  the  graph  of  35!; +3. 

The  distances  that  are  represented  by  the  verticals  might 
have  been  found  by  substituting  —4,  —3,  —2,  —  i,  o,  +1, 
+  2,  +3,  +4  in  turn  for  x,  calculating  the  corresponding 
values  -9,  -6,  -3,  o,  +3,  +6,  +9.  +12,  +15,  of  3:^+3, 
and  then  representing  these  numbers  to  scale  on  the  verticals. 
Graphs  may  be  drawn  for  any  expressions  in  the  same 
way. 

1.  Draw  a  graph  of  3/— 3;  of  2/4-3;  o^  2/— 3;  of  3/4-2; 
of  2/  — 2. 

2.  Draw  graphs  of  the  following  expressions: 

(i)  2a;+i  (4)  3^-4  (7)  2^-1-4 

(2)  2:Jf-i  (5)  ^-2,  (8)  4^-1-2 

(3)  3^+4  (6)  4^+3  (9)  2X-/^. 

150.  The  graphs  show: 

(i)  that  there  are  a  great  many  values  of  any  one  of  these 
expressions,  one,  indeed,  for  every  value  of  /,  or  of  x,  we  care 
to  substitute; 

(2)  that  there  is  but  one  place,  or  point,  where  the  value 
of  the  expression,  as  331; +3,  is  o; 

(3)  that  the  value  of  x  that  makes  3^-}- 3,  or  other  like 
expression,  o,  is  the  distance  from  O  in  the  graph  to  the 
crossing-point  of  the  line  of  points  and  the  horizontal  line. 

This  same  value  of  x  that  makes  3:x;4-3  equal  to  o  can  be 

more  quickly  found  by  solving  the  equation,  3^4-3=0,  for  x, 

thus: 

3x+3=o 

3« 3 

«=— I 
Check:  3  •  (-i)-<-3= -34-3=0. 

151.  When  any  equation  as  3^1;— 6=0  is  written  down, 
it  may  be  regarded  as  merely  symbolizing  the  question: 


The  Simple  Equation  in  One  Unknown  2 1 3 

What  value,  or  what  values,  if  any,  of  x  are  there,  which 
substituted  in  ^x—6  will  make  3:^:— 6  equal  to  ol 
Two  kinds  of  equation  are  used  in  algebra: 

I.  The  identical  equation,  or  the  identity,  is  an  equation 
that  is  true  for  any,  or  all,  values  of  the  letters. 

Examples:  i.  (x  +  ^){x—i)=x^  —  g. 

2.  {x  +  a)(x  +  a)=x^  +  2ax  +  a'. 

Test  whether  example  i  is  true  for  x=4;  for  x  =  'j;  for 
:x;  =  io;  for  x  =  i. 

Test  whether  example  2  is  true  for  x=^,  a=i;  for:x;  =  5, 
a  =—2;  fora:  =  i2,  a  =  io. 

Identical  equations  are  sometimes  Written  with  the  sign  =, 
when  it  is  desired  to  distinguish  them  from  other  kinds  of 
equations. 

II.  The  conditional  equation  is  an  equation  that  is  true  only 
for  one  or  for  a  definite  number  of  values  of  its  letters. 

Examples:   i.  5:x;— 15  =  25. 

2.  3i;»  — 16  =  0. 

3.  {x-l)(x+2){x-7,)=0. 

Test  whether  example  i  is  correct  for  a; =8;  for  any  other 
value  of  X. 

Test  whether  example  2  is  correct  for  x=4;  for  a;  =—4; 
for  any  other  values  of  x. 

Test  whether  example  3  is  correct  for  51;  =  i;  ior  x=—2; 
for  x=3;  for  any  other  values  of  :x;. 

When  nothing  is  said  to  the  contrary,  the  word  "equation" 
means  conditional  equation. 

The  left  side  of  an  equation  is  called  the  left  member,  or  the 
first  member,  and  the  right  side,  the  right  member,  or  the 
second  member. 

Point  out  which  of  the  following  equations  are  (i)  identical 
equations,  or  identities,  (2)  conditional  equations,  and  give 
reasons  for  your  answers: 


214  First- Year  Mathematics 

1.  3C-i=o  lo.  a{b-s)=ab-sa 

2.  x'—4=o  II.  {a+x){a—x)=a'—x' 

3.  (x-2y=x'-^-\-4  12.  (a-3)(fl+4)=o 

4.  x'-i6  =  {x+4){x-4)  13.  (a-3)(a+4)  =«="+»- 12 

5.  a{x-s)=ax-:ia  c+8    c  , 

6.  a(3f-3)=a  .^44 

7.  ic(:x;— i)=:!c^— 31;  (,-—7 

8.  ic(5(;  — i)=o  -^      5 

9.  :s;3_27=o  16.  m'^-n*  =  (m'+n'){tn'~n'). 

While  any  letter  may  be  used  to  denote  the  primary  unknown 
(or  primary  variable),  it  is  customary  to  use  x,  or  y,  or  z,  or 
some  one  of  the  later  letters  of  the  alphabet  for  this  purpose. 

The  earlier  letters  of  the  alphabet,  as  a,  b,  m,  I,  k,  etc.,  are 
quite  commonly  used  to  denote  fixed,  or  constant,  numbers, 
like  6,  15,  75,  etc.  The  letters  that  abbreviate  words,  as  i  for 
interest,  b  for  base,  are  also  much  used  to  denote  numbers. 

Stating  Algebraic  Problems 

152.  The  problems  of  the  foregoing  chapters  of  this  book, 
as  well  as  those  of  modern  arithmetic,  show  that  letters  may 
be  used  to  advantage  to  express  numbers,  and  to  simplify 
stating  and  solving  problems.  Algebraic  skill  means  nearly 
the  same  thing  as  skill  in  stating  and  solving  problems.  This 
skill  can  be  acquired  only  through  much  practice,  and  not  by 
committing  rules  and  directions.  Still,  it  may  aid  to  know 
that  in  algebra  stating  a  problem  usually  means  expressing  in 
the  language  of  algebraic  symbols  certain  number-relations 
that  are  given  in  verbal  language. 

The  statement  of  a  problem  in  most  cases  takes  the  form 
of  an  equation.  To  obtain  this  equation,  the  following  may 
serve  to  guide  the  beginner: 

I.  Denote  the  unknown  number  by  a  letter,  then  translate 
the  verbal  statement  of  the  number-relations  into  a  symbolic 
statement  in  equation  form. 


The  Simple  Equation  in  One  Unknown  215 

The  first  letters  of  words  are  convenient  letters  to  use  to 
denote  numbers,  while  they  are  yet  unknown;  as  n  for  num- 
ber, t  for  time,  a  for  age,  etc. 

Give  statements  of  the  following  problems,  then  solve  and 
check  them. 

1.  Two-thirds  of  a  number  diminished  by  12  equals  4. 
Find  the  number. 

Statement: 

Two-thirds  of  a  number  diminished  by  1 2  equals  4. 
§  X         n  —  12     =      4. 

Solve  the  equation  and  check  by  substitution. 

2.  Twice  a  number  decreased  by  12  is  the  same  as  the 
number  increased  by  f  of  itself.     Find  the  number. 

3.  Six  times  a  number  increased  by  ^  of  itself  equals  11. 
Find  the  number. 

4.  Four  times  a  number  increased  by  ^  of  itself  is  the 
same  as  twice  the  number  increased  by  11.  Find  the 
number. 

5.  The" acute  angles  of  a  right  triangle  are  /^x  and  ^x.  Find 
the  acute  angles. 

6.  One  of  two  supplementary  adjacent  angles  is  3  times 
the  other.     Find  the  two  angles. 

7.  The  angles  for  a  triangle  are  x,  2X,  and  3^1;  degrees. 
Find  the  numerical  values. 

Another  guide  that  is  often  useful  in  stating  a  verbal 
problem  in  equation  form  is  the  following: 

II.  By  the  aid  of  literal  and  of  arithmetical  numbers  express 
some  number  of  the  problem  in  two  different  forms,  and  write 
the  two  expressions  equal  to  one  another. 

Thus  an  equation  is  a  statement,  in  symbols,  that  two 
different  number-expressions  stand  for  the  same  number. 

8.  A  rectangle  is  3  times  as  long  as  wide,  and  the  perim- 
eter is  48  inches.     Find  the  width  and  length. 

Equate  two  different  expressions  of  the  perimeter. 


2 1 6  First-  Year  Mathematics 

9.  Double  the  number  of  years  in  a  boy's  age  is  16  more 
than  his  age  2  years  ago.     How  old  is  the  boy  ? 

Equate  two  different  expressions  for  the  difference  of  double  the 
boy's  present  age  and  his  age  two  years  ago. 

10.  The  difference  of  double  a  boy's  age  and  3  times  his 
age  10  years  ago  is  his  present  age.     How  old  is  the  boy  ? 

STATEME^r^  in  Symbols: 

2x—2,{x—jo)      =  X  (l) 

—3(«— 10)  =—3a(;  +  30,  hence  2x—3»+3o=  x  (2) 

2x—T,x  =—x,  hence  — x+30       =  x  (3) 

Adding  x  to  both  sides  30  =  2X  (4) 

Dividing  by  2  15  =  ».  (5) 

Check  by  testing  in  the  conditions  of  the  problem. 

The  boy's  age  is  15  years. 

153.  The  solution  of  an  equation  consists  in  getting  another 
equation  in  which  the  unknown  stands  alone  in  one  member, 
with  the  numbers  in  terms  of  which  the  unknown  is  to  be 
expressed,  in  the  other  member.  The  required  changes  are 
made  by  means  of  the  axioms  stated  on  p.  26,  §21,  or  by 
means  of  the  fifth  axiom  now  added  below.  The  five  axioms 
that  are  to  be  the  reasons  for  changes  in  equations,  are  now 
stated  in  final  form. 

I.  If  the  same  number,  or  eqtial  numbers,  be  added  to  equals 
the  sums  are  equal.     (Addition-axiom.) 

11.  If  the  same  number,  or  equxil  numbers,  be  subtracted 
from  equal  numbers,  the  remainders  are  equal.  {Subtraction- 
axiom.) 

III.  //  equal  numbers  be  multiplied  by  the  same  number,  or 
by  equal  numbers,  the  products  are  equal.  (Multiplication- 
axiom.) 

IV.  If  equal  numbers  be  divided  by  the  same  number,  or  by 
equal  numbers,  the  quotients  are  equal.     (Division-axiom.) 

Division  by  zero,  or  by  any  expression  in  which  the  letters  have 
values  that  make  the  divisor  zero,  is  not  permitted. 


The  Simple  Equation  in  One  Unknown  217 

V.  Any  change  may  he  made  in  the  form  of  an  expression, 
that  does  not  change  its  value. 

One  of  the  most  important  form-changes  is  the  removal 
and  introduction  of  parentheses. 

When  a  reason  for  a  change  is  asked,  the  pupil  should  be 
able  to  quote,  or  to  cite,  the  axiom  that  justifies  the  change. 

154.  Give  the  reason  for  the  truth  of  the  conclusion  in 
each  of  the  following: 

1.  a  =9   and  &=3,  thena+&  =  i2 

2.  w=i3  and  w=2,  then    mn  =  26 

3.  a  =x  Sind  b=y,  then  a-\-b=x+y 

4.  c  =16  and  d=4,  then  c—d  =  i2 

m 
K.  w  =  28  and  w  =  7,    then  — =4 

^     c  ,   d  ,       c+d     „ 

0.  —  =  5  and — =3,   then =8. 

12  12  12 

Give  reasons  for  the  following: 

7.  If  a  =  7,         then  5^=35      11.  If  5^—5=3,    then  x =8 

/  r 

8.  If -^=3,       then  7=48        12.  If— =  7,        then  r  =  70 

10  10 

g.  li  mn—6n,    then  w =6  xr  ,  .  b 

^.  o      ,,  13-  li  ax  =  b,        then5(;=-' 

10.  If  23^  =  18,     then  y  =  g  a 

155.  To  solve  an  equation  means  to  find  the  number,  or 
the  numbers,  which,  substituted  for  the  unknown  in  the  equation, 
reduces  both  members  to  the  same  number. 

Any  number  that  fulfills  these  conditions  is  a  root  of  the 
equation. 

The  solution  of  an  equation  means  two  different  things  in 
mathematics,  viz.: 

(i)  The  solving  of  the  equation. 

(2)  The  value,  or  values,  found  by  solving  the  equation. 


2i8  First- Year  Mathematics 

In  this  book  the  word  "solution,"  from  now  on,  usually 
has  the  second  meaning. 

To  check  tJie  process  of  solving  an  eqwition  means  to  show 
that  the  work  oi  finding  the  unknown  is  correct. 

This  is  often  done  by  checking  the  "solution,"  i.e.,  the 
result,  though  it  is  generally  desirable  to  examine  the  reasoning, 
or  to  use  an  independent  process. 

Distinguish  between  root  of  a  number  (pp.  24  and  28)  and 
root  of  an  equation. 

To  check  the  solution  of  an  equation  means  to  show  that 
the  result  is  a  root  of  the  equation.  This  is  commonly  done 
by  substituting. 

To  check  the  solution  of  a  problem  means  to  show  that  the 
result  answers  the  conditions  stated  in  the  problem.  To  do 
this  the  result  must  be  tested  by  the  stated  conditions,  and  not 
merely  by  substituting  in  the  equation  solved.  This  equation 
may  itself  be  wrong. 

Exercises  for  Practice 

156.  It  should  be  borne  in  mind  that  finding  the  root  of 
an  equation  includes  checking  the  result. 
Find  the  roots  of  the  following  equations: 
I.  sx+g  =  :ix  +  i'j. 

Full  Solution: 

5^  +  9=3^+17 
Subtract  3»  33:       =^x  (by  Axiom  II) 

2;X  +  9=  17 

Subtract  9  9=9  (Ax.  II) 

2X        ^-  8 

Divide  by  2  x       =  4  (Ax.  IV) 

Check:  $x+  9  =  5-4+  9  =  29 

3^+i7=3-4  +  i7  =  29- 
Hence,  4  is  a  root  of  5^+9  =  30;+ 17. 


The  Simple  Equation  in  One  Unknown  2 1  g 

Shortened  Solution: 

5^  +  9  =  3^+17 
Subtract  2)X-\-g  from  both  sides  (Ax.  II)  getting 

x  =  ^. 
Check:  5  •4  +  9  =  29=3  •4  +  17. 
Hence,  4  is  a  root  of  5ar+9  =  3:x;  +  i7. 

Note:  The  reasons  need  not  be  written,  but  they  should  be  thought. 

In  practice  only  this  is  needed: 

5^+9  =  3^+17 
2^  =  8 
x  =  ^ 
Check:  5.4  +  9  =  29  =  3.4  +  17. 

2,   <)X-\-i']=^x  —  2  16.  2>{x  —  2)=']X  —  g 

3-  3^+5  =  5^-15  17.  ii-2,{x-2)^x-?> 


4.  6x—']=iQX-]r'i^ 


2>x 


5.  2:x;— 3=3^  —  7  18.  —+5—^  +  3 


6.  2>x-\-iS=x-\-2S 

7.  9X  — 8  =  25— 2X 

8.  i(>-\-T)X=6-\-x 


19.  35f  — 2(ic  +  5)=6x  — 20 

20.  3(:>f-2)  +  i5  =  5x-3 


9.  20-4.^=8-10^  21.  ^+5  =  91 -lox 

10.  18— 5(;=4JC+3  4 

11.  55f-i7+3X-5=x-i.  22.  5(5f-i)=3(5(;  +  i) 

ByAxiomV,8.-22=.-x  8(3-2:.) -2(5-^)  =28 

12.  65c— 7— 83(;  +  ii5=o 

o  1         o  24.     —  ^X  — 24  =  '?-?(2— ^c) 

13.  8«;  — 22  =  — 23f  +  io8  tot    oov        / 

14.  7.^-9-io;c  =  3.r-i5  25.  5(i-i3^)=35-io5x 

15.  — l-2:x;— 8  =  3x  — 5  26.  8  +  2:x;H —  =  i|H 

4  4  3 

27.  3ac  +  i4-5(3(;-3)=4(ac+3) 

28.  2(x-i)+3(x-2)=4(:x;  +  3) 

29.  2(:x;  +  i) +3(^  +  2) +4(5(;  +  3)  =110 

30.  5(x-2)+6(x-i) -4(^-5)  =60 

2X  +  -]      X  —  2       7 

31- = 

5  3        5 


220  First-Year  Mathematics 


32.  ^ ^-8^, 


33.  x{x-i)-x{x-2)=4{x-s) 
13(^-5) 


34.    ^^^^ ^H =  2X 

4  5 


35.  4(2«;+9) +3(^-9)  =  - 

Problems  Leading  to  Equations  in  One  Unknown 
157.  State  and  solve  the  following  problems  and  interpret 
the  results: 

1.  A  rectangle  is  2  units  longer  than  three  times  the  width, 
and  the  perimeter  is  60.     Find  the  length  and  the  width. 

2.  The  area  of  a  triangle,  whose  altitude  is  5  units  less 
than  the  base,  is  equal  to  10  square  units  less  than  half  the 
area  of  a  square  on  the  base  of  the  triangle.  Find  the  base 
and  the  altitude  of  the  triangle. 

3.  One-fourth  of  the  difference  of  3  times  a  number  and 
8  is  10.     Find  the  number. 

4.  A  is  160  yd.  east  and  B  112  yd.  west  of  a  gate.  Both 
start  at  the  same  time  to  walk  toward  the  gate,  A  going  3  yd. 
and  B  2  yd.  a  second.  When  will  they  be  at  equal  distances 
from  the  gate  ? 

5.  Baking  powder  is  made  up  of  as  much  starch  as  soda 
and  twice  as  much  cream  of  tartar  as  soda.  How  much  of 
each  ingredient  is  there  in  2  lb.  of  baking  powder  ? 

6.  An  express  train  whose  rate  is  40  mi.  per  hour,  starts 

I  hr.  4  min.  after  a  freight  train,  and  overtakes  it  in  i  hr. 

36  min.     How  many  miles  per  hour  does  the  freight  train  nm  ? 

Let  some  letter,  as  x,  be  the  rate  of  the  freight  train  and  notice  that 
the  trains  go  the  same  distance. 

7.  Two  trains  start  at  the  same  time  from  S,  one  going 
east  at  the  rate  of  35  miles  per  hour,  and  the  other  going  west 


The  Simple  Equation  in  One  Unknown  221 

at  a  rate  \  faster.     How  long  after  starting  will  they  be 
exactly  100  miles  apart? 

8.  Sound  travels  1,080  ft.  per  second.  If  the  sound  of  a 
stroke  of  lightning  is  heard  3^  seconds  after  the  flash,  how  far 
away  is  the  stroke  ? 

9.  If  sound  travels  /  ft.  per  second,  how  far  away  is  a 
lightning  stroke  if  the  sound  is  heard  5  seconds  after  the  flash  ? 

10.  A  tree  half  a  mile  distant  was  struck  by  lightning. 
It  took  2 1  seconds  for  the  sound  to  reach  the  ear.  Find  the 
rate  in  feet  per  second,  at  which  the  sound  traveled. 

11.  A  tree  m  feet  distant  was  struck  by  lightning.  It 
took  t  seconds  for  the  sound  to  arrive.  What  was  the  rate 
in  feet  per  second  at  which  the  sound  traveled  ? 

12.  A  man  walks  beside  a  railway  at  the  rate  of  4  mi. 
per  hour;  a  train  208  yd.  long,  running  30  mi.  per  hour, 
overtakes  him.  How  long  will  it  take  the  train  to  pass  the 
man? 

13.  A  man  walks  beside  a  railway  at  the  rate  of  m  mi. 
per  hour.  If  a  train  /  yd.  long  and  traveling  n  mi.  per  hour 
overtakes  him,  how  long  will  it  take  the  train  to  pass  the  man  ? 

14.  Solve  the  following  exercises,  as  many  as  possible 
mentally : 

(1)  ^+6=8  (8)  3:,-i:^=-7 

,  .    2r  ,  .  4^  —  3 

(2)  —-3  =  1  (9)  12 — -=x+3 

3  7 

(3)  105—8=3—5  (10)  ax—hx=(ia—6h 

(4)  3^  +  13-5^=^  (11)  3«^+^  =  i2a+4 

(O  3^-5  (12)  %-{s-2x)  =  -x 

^^^       2        ^  ,    ,  x-a 

(13)  ax —  =a=' 


(6)  10— :x;=— 20  c 

~  3 


,  .   351;— 2  .  ,                       ,    .           a(x—c)     sac 
(7) h6  =  2A;  (14)  ax — ^^ -^ 


aaa  First-Year  Mathematics 

15.  A  train  moves  at  a  uniform  rate.  If  the  rate  were 
6  mi.  per  hour  faster  the  distance  it  would  go  in  8  hr.  is  50  mi. 
greater  than  the  distance  it  would  go  in  11  hr.  at  a  rate  7  mi. 
per  hour  less  than  the  actual  rate.  Find  the  actual  rate  of 
the  train. 

16.  Two  trains  go  from  P  and  Q  on  different  routes,  one 
of  which  is  15  mi.  longer  than  the  other.  The  train  on  the 
shorter  route  takes  6  hr.,  and  the  train  on  the  longer,  running 
10  mi.  less  per  hotu:,  takes  2  J  hr.  Find  the  length  of  each 
route. 

17.  The  distance  from  A  to  B  is  100  mi.  A  train  leaving 
A  at  a  certain  rate,  meets  with  an  accident  20  mi.  from  B, 
reducing  the  speed  one-half  and  causing  it  to  reach  B  i  hr. 
late.    What  was  the  rate  per  hour  before  the  accident  ? 

18.  An  express  train  whose  rate  is  r  miles  per  hour,  starts 
h  hours  after  a  freight  train  and  overtakes  it  in  t  hours.  Find 
the  rate  per  hour  of  the  freight  train. 

19.  Two  trains  start  from  the  same  place  at  the  same  time, 
one  going  east  at  the  rate  of  m  miles  per  hour,  the  other  going 
west  at  the  rate  of  n  miles  per  hour.  How  long  after  starting 
will  they  be  c  miles  apart  ? 

20.  A  man  rows  downstream  at  the  rate  of  6  miles  per 
hour  and  returns  at  the  rate  of  3  miles  per  hour.  How  far 
downstream  can  he  go  and  return  in  9  hours  ? 

21.  At  what  time  between  3  and  4  o'clock  are  the  hands 
of  the  clock  together  ? 

Let  X  denote  the  number  of  minute-spaces  over  which  the  minute 
hand  passes  from  3  o'clock  until  it  first  overtakes  the  hour  hand,  then 

show  that 1-15=3:,  whence,  x  =  i6^r. 

Hence,  the  hands  are  together  at  161^  minutes  past  3  o'clock. 

22.  At  what  time  between  2  and  3  o'clock  are  the  hands 
of  the  clock  together? 


The  Simple  Equation  in  One  Unknown 


223 


23.  At  what  time  between  3  and  4  o'clock  are  the  hands 
of  the  clock  at  right  angles  (two  results)  ? 

24.  At  what  time  between  7  and  8  o'clock  are  the  hands 
of  the  clock  pointing  in  opposite  directions  ? 

25.  Solve  the  following  for  x: 

(^)  a+3  — 

XX       I 

^'^  -a-^b=ab 

(3)  -7>{x-\-a)  =  2{a-7,x) 

(4)  a'x—b'x=a—b 

(5)  b^x—a'x=a—b 

(6)  ^--3+e^^=6 

15  4 


(8)  -;+^=f +4 
340 

(9)  -j---  =  ^^ 

2       4 

(10)  — a;=— cH — 
c  c 

,      .  2  —  2X  , 

(11)  5:x:— ^^^ =2X-\-2ii 

/     X         d—x  , 

(12)  3!; — —  =  2-{x-^) 


(7)  ^- 


(13)  cx- 


a^{x—^)     ^a^    a—c 


'XmrU 


c  c         c 

26.  Venus  makes  its  orbit  in  224.7  days  or  in  about  7^ 
months ;  the  earth  starting  as  shown 
in  the  Fig.  225.     In  how  many  days 
will  Venus  next  be  in  line  between 
the  earth  and  the  sun? 

The  rate  per  month  of  Venus  is  y\ 
of  the  orbit,  that  of  the  earth  ^.  Let  x 
be  the  required  number  of  months. 

Express  by  an  equation  that  Venus 
must  make  one  more  revolution  than  the 
earth. 

27.  CaUing  the  revolution  of  Venus  about  the  sun  7^ 
months,  and  that  of  Mercury  3  months,  how  many  months 
after  Mercury  is  in  the  line  between  Venus  and  the  stm  will 
it  next  be  in  the  same  relative  position  ? 

28.  Seen  from  the  earth,  the  moon  completes  the  circuit  of 
the  heavens  in  about  27  days,  8  hours,  and  the  sun  in  365 


Fig.  225 


224  First-Year  Mathematics 

days,  6  hours,  in  the  same  direction.  Required  the  time,  to 
.001  day,  from  one  full  moon  to  the  next,  the  motion  supposed 
uniform. 

29.  If  a  revolution  of  one  planet  is  a  days  and  of  another 
h  days  (a<b),  how  many  days  are  there  between  successive 
similar  relative  positions  ? 

30.  Use  the  result  of  29  as  a  formula,  and  solve  problems 
25  to  27  by  means  of  it. 

IT 

31.  Using  the  formula,  t^ir-^-  ,  find  how  many  times  as 

fast  as  a  2-foot  pendulum  a  6-inch  pendulum  vibrates. 

32.  The  difference  of  the  squares  of  two  consecutive  nimi- 
bers  is  19.     Find  the  numbers. 

33.  The  difference  of  the  squares  of  two  consecutive  num- 
bers is  273.     Find  the  numbers. 

34.  The  difference  of  the  squares  of  two  consecutive  num- 
bers is  a.     Find  the  numbers. 

35.  Solve  problems  32  and  33,  using  the  result  of  34  as  a 
formula. 

36.  The  difference  of  the  squares  of  two  consecutive  even 
numbers  is  28.     Find  the  numbers. 

37.  The  difference  of  the  squares  of  two  consecutive  even 
numbers  is  100.     Find  the  numbers. 

38.  The  difference  of  the  squares  of  two  consecutive  even 
numbers  is  a.     Find  the  numbers. 

39.  Solve  problems  36  and  37  using  the  result  of  38  as  a 
formula. 

40.  The  difference  of  the  squares  of  two  consecutive  odd 
numbers  is  48.     Find  the  numbers. 

41.  The  difference  of  the  squares  of  two  consecutive  odd 
numbers  is  5.     Find  the  numbers. 


The  Simple  Equation  in  One  Unknown  225 

42.  Using  the  result  of  problem  41  as  a  formula,  find  two 
consecutive  odd  numbers  the  difference  of  whose  squares  is  56. 

43.  Solve  the  following  equations  (i)  for  x;  and  when 
possible,  (2)  for  a;  (3)  for  z: 

(i)  x-{-a  =  2a—z  ^_i_^_ 

(2)  T,x  —  2a=z—a  X    X 

a      z  ...  6x-\-i     x-i 

(3>i-i5-»  (8)  15^— --  — =-6 

(4)  ±^^=,  (,)  _?+^  =,_._'_ 

ax    xz  6(x— i)     ;x;— I  x  —  i 

U)  axz  =  i2  ,    .   X    6(x  — i)     \—2X       - 

y'  ^  (10) ^ ~ =2*- 

(6)  axz=(iX-\  '    '   3  5  3 

44.  Bell-metal  is  by  weight  5  parts  tin  and  16  parts  copper. 
How  many  pounds  of  tin  and  copper  are  there  in  a  bell  weigh-, 
ing  4,800  pounds  ? 

First  Method:   Let  »  be  the  number  of  pounds  of  tin,  then  show 

that  x-\ =4,800,  and  solve  for  x,  and  find  • —  . 

5  5 

Second  Method  (without  fractions):  Let  5a;  be  the  number  of 
pounds  of  tin,  and  solve:    551;+ i6:x;  =  4,800. 

After  finding  ac,  calculate  5:!C  and  i6x. 

45.  Gunpowder  contains,  by  weight,  6  parts  saltpeter, 
I  part  sulphur,  and  i  part  charcoal.  How  many  pounds  of 
saltpeter,  of  sulphur,  and  of  charcoal  are  there  in  120  pounds 
of  gunpowder  ? 

46.  If  gunpowder  were  composed  of  4  parts,  by  weight, 
of  saltpeter,  2  parts  sulphur,  and  3  parts  charcoal,  how  many 
pounds  of  each  would  there  be  in  200  pounds  of  gunpowder  ? 

47.  With  ingredients  as  in  problem  45,  how  much  salt- 
peter is  burned  in  the  discharge  of  a  cannon  using  50  pounds 
of  powder  to  the  cartridge  ? 

48.  Baking  powder  is  composed  of  4  parts,  by  weight,  of 
cream  of  tartar,  i  part  starch,  and  i  part  soda.  How  much 
of  each  ingredient  is  there  in  18  lb.  of  baking  powder  ? 


226  Fir  St- Year  Mathematics 

49.  A  certain  mixture  weighing  m  pounds  contains  a  parts 
by  weight  of  copper,  b  parts  of  iron,  and  c  parts  of  carbon. 
How  many  pounds  of  each  ingredient  are  there  in  the  mixture  ? 

50.  Solve  45,  46,  47,  and  48,  using  the  result  of  49  as  a 
formula. 

51.  In  a  watch-case  weighing  2  ounces,  the  gold  is  14 
carats  fine,  i.  e.,  there  are  14  parts  of  gold  in  every  24  parts 
of  the  whole  alloy.  How  many  ounces  of  pure  gold  are  there 
in  the  case  ? 

52.  In  an  alloy  of  silver  and  copper  weighing  90  ounces, 

there  are  6  oimces  of  copper.     Find  how  much  silver  must  be 

added  so  that  10  ounces  of  the  new  alloy  shall  contain  f  ounce 

of  copper. 

Let'a;  be  the  number  of  lb.  of  silver  added.  The  new  compound 
then  weighs  (qo  +  ^x;)  lb. 

53.  A  certain  compound  contains,  by  weight,  5  parts  car- 
bon to  every  3  parts  of  iron,  and  7  parts  of  iron  to  every  2  parts 
of  copper.  In  1 24  pounds  of  the  compound,  how  many  pounds 
are  there  of  carbon,  of  iron,  and  of  copper  ? 

54.  If  80  pounds  of  sea-water  contains  4  pounds  of  salt, 
how  much  fresh  water  must  be  added  to  make  a  new  solution 
of  which  45  pounds  contain  f  pound  of  salt  ? 

55.  In  a  mass  of  alloy  for  watch-cases,  which  contains 
60  oz.,  there  are  20  oz.  of  gold.  How  much  copper  must  be 
added  so  that  in  a  case  weighing  2  oz.  there  will  be  ^  oz. 
of  gold  ? 

56.  In  an  alloy  weighing  80  grams,  there  are  34  grams  of 
gold.  How  much  nickel  must  be  added  so  that  a  ring  made 
from  the  new  alloy  and  weighing  i|  grams  shall  contain  J  gram 
of  gold  ? 

57.  In  an  alloy  weighing  a  oz.  there  are  b  oz.  of  gold. 
How  much  of  another  metal  must  be  added  so  that  a  portion 
weighing  c  oz.  shall  contain  d  oz.  of  gold  ? 


The  Simple  Equation  in  One  Unknown  227 

58.  Solve  the  following  for  5: 

(i)  s{s-i)-s{s-2)=2> 

(2)  (5  +  l)(5  +  3)  =  (5  +  2)(5  +  S)-13 

(3)  4(^+6) -2(5-3)  =38 

(4)  (^+3)  =^+3(^+8) 
s  —  i      s-\r^ 


(5) 


5  +  1       5  +  17 


(6)  ^I±l-^+l^o 

II  ?> 


(7) 
(8) 


(5  +  3)(5  +  6)       (5  +  2)(5  +  8) 

I       _3__^ 4_ 

5—1     5—3    5—2    5—4 


,   s    5-5       5-6^5+3      5+4 
5—3      5—4      5+1       5+2 

/     N  -^-4  .  5  +  4     5  +  20     5-5 

(10) 1 == 1 

3         7         10         5 

5-1       5-2 

^4  22 

59.  A  pound  of  lead  loses  -^  of  a  pound,  and  a  pound  of 

iron  loses  ^  of  a  pound  when  weighed  in  water.     How  many 

pounds  of  lead  and  of  iron  are  there  in  a  mass  of  lead  and 

iron  that  weighs  159  lb.  in  air  and  143  lb.  in  water? 

If  an  object  weighing  2  lb.  in  the  air  is  suspended  by  a  cord  and 
weighed  when  immersed  in  water,  it  will  weigh  less  than  2  lb.  It  can 
be  shown  that  the  loss  of  weight  is  the  same  as  the  weight  of  the  water 
the  object  displaces. 

60.  A  mass  of  gold  weighs  97  oz.  in  air  and  92  oz.  in 
water,  and  a  mass  of  silver  weighs  21  oz.  in  air  and  19  oz.  in 
water.    How  many  ounces  of  gold  and  of  silver  are  there  in 


228  First- Year  Mathematics 

a  mass  of  gold  and  silver,  that  weighs  320  oz.  in  air  and  298 
oz.  in  water  ? 

61.  Into  what  two  sums  can  $1,000  be  divided  so  that  the 
income  of  the  one  at  6%  shall  equal  the  income  of  the  other 
at4%?     . 

62.  How  can  a  man  divide  $2,000  so  that  the  income  of 
part  at  4%  shall  be  the  same  as  that  of  the  rest  at  5%  ? 

63.  How  many  dollars  must  be  invested  at  4%  to  give 
the  same  income  as  that  of  $2,500  at  6%  ? 

64.  What  per  cent,  of  evaporation  must  take  place  from 
a  6%  solution  of  salt  and  water  (salt-water  of  which  6%  by 
weight  is  salt)  to  make  the  remaining  portion  of  the  mixture 
an  8^  solution?    A  13%  solution? 

Let  X  be  the  number  of  per  cent,  evaporated,  then  100— :» is  the  per 
cent,  remaining.     Then  show  .08(100— a;)  =  .06X100. 

65.  What  per  cent,  of  evaporation  must  take  place  from  a 
90%  solution  to  produce  a  95%  solution  ? 

66.  What  per  cent,  of  evaporation  must  take  place  from  aii 
a%  solution  to  produce  a  b%  solution?  {b>a). 

67.  A  physician  having  a  6%  solution  of  a  certain  kind  of 
nedicine  wishes  to  dilute  it  to  a  3^%  solution.     What  per  cent. 

ot  water  must  be  added  ? 

68.  A  druggist  has  a  95%  solution.  What  must  he  do  to 
change  it  to  an  80%  solution  required  in  a  prescription  ? 

69.  From  a  mixture  of  sugar  and  water  of  30%  strength. 
\  of  it  evaporated  (removing  water  only).  What  per  cent,  of 
strength  was  the  remainder  ? 

70.  From  a  mixture  of  medicine  and  alcohol  of  28% 
strength,  ^  was  evaporated  (removing  alcohol  only).  What 
per  cent,  of  strength  was  the  remaining  medicine  ? 


The  Simple  Equation  in  One  Unknown  229 

71.  Solve  the  following  equations  for  r: 

(i)  o.25r-o.j25r=o.75 

,      r+2.i  r 

(2)  — +— =  2.275 

r+o.i     r+0.7  '^ 

(3) '- — ^"-i 

.      a-\-r     a—r  __a—r 
a+c     c—a     c—a 

(5)  5^-13=—-^+—^ 
4  4 

5y''-3r  +  i2^^^(3r  +  i)(r-io) 

7  42 

(7)  =^ 6 — +^^=-' 

6r  +  i     r  — I 
(8)  15--^ 3—6. 

72.  A  box  of  oranges  was  bought  at  the  rate  of  15  cents  a 
dozen.  Five  dozen  were  given  away  and  the  remainder  sold 
at  the  rate  of  2  for  5  cents.  If  this  gave  a  profit  of  30  cents  on 
the  box,  how  many  were  there  in  the  box  ? 

73.  A  father  engaged  his  son  to  work  20  days  on  the  fol- 
lowing conditions:  For  each  day  he  worked  he  was  to  re- 
ceive $2,  and  for  each  day  he  was  idle  he  was  to  forfeit  $1. 
At  the  end  of  20  days  he  received  $34.  How  many  days 
was  he  idle  ? 

74.  A  man  engaged  to  work  on  the  following  conditions: 
For  each  day  he  worked  he  was  to  receive  b  dollars,  and  for 
each  day  he  was  idle  he  was  to  forfeit  c  dollars.  At  the  end 
of  a  days  he  received  d  dollars.  How  many  days  was  he 
idle? 


230  First-Year  Mathematics 

75.  The  tire  of  the  fore- wheel  of  a  carriage  is  9  ft.,  that 
of  th3  hind-wheel  12  ft.  What  distance  will  the  carriage  have 
passed  over  when  the  fore-wheels  have  made  5  more  revolu- 
tions than  the  hind-wheels  ? 

76.  The  tire  of  the  fore-wheel  of  a  carriage  is  a  ft.,  that 
of  the  hind-wheel  b  feet.  What  distance  will  the  carriage 
have  passed  over  when  the  fore-wheels  have  made  n  more 
revolutions  than  the  hind-wheels  ? 

77.  A  man  spends  one  ath  part  of  his  income  for  food,  one 
6th  part  for  rent,  one  c\h  part  for  clothing,  one  dih  part  for 
furniture,  and  he  saves  e  dollars.     How  much  is  his  income  ? 

Exercise  XXXHI 
158.  Solve  the  following  equations: 

1.  A  = h,  for  h;  for  h;    or  h' 

2 

Prt 

2.  i= —  for  p\  for  r ;  for  /;  for  pr\  for  rt 

3.  V  =  \hh  for  6;  for  A 

4.  C  =  K^-32)fori? 

5.  pd=PD,  for  p;ioTd;ioxP;  for  D 

6.  C=r=——  for  E;  for  R;  for  r 

R+r 

7.  p=Y+J  ^^^  ^;  for/i;    for  /, 

8.  C=^-i-^forC,;  f or  C, 

9.  iTFL=—  for  W;  for  S;  for  - 

c  c 

10.  'z;='yo(iH )  forz/o',  for/. 

\      273/ 


The  Simple  Equation  in  One  Unknown  23 1 

Exercise  XXXIV 
159.  Solve  the  following  exercises  for  x: 

1.  ax-bx=-{b-a)  11.  x3-2ax-\-c=a-\-^ax-\-x3 

2.  cx-\-ax=a''+c^  +  2ac  12.  a''+ax+cx—ac=o 

3.  ^x-ax=a''+g-6a  13.  yx-2cm  =  yx—2rm 

4.  fn^x—n'x=n+m  2X—I    x  —  2s 

5.  5^oc+r^3C  — 2r5:x;=r^— 5'  ^'*'  6;x;+5~'35(;+6 
3C_5C^6^— a^*  251;— a    jc— a    a 


ah        ab  ^^' 


c 


X      ag_6a— 86  c:x;— rf    dx—c         c—d 

4&    3^      I2a6  ■     Jx  c^e  cdx 

„   X  I  I     2  2.r+a    «;+a        a 

8.  — «;  =  i+— —  17. —  =  — 

a  a""    a  '        c  d  c 

9.  i2X-i){T,-x)  =  -2{x''-g)  x^'-d    d+x_2X    d 
10.  x'—c'  =  s(^^  —  2cx+x''                'ex  c         ex 

19.  {x-\-my  -\-{n+sy  =  {x-my  ^-{n-sy 
r—s     s—t_r—t 
x—t    x—r      X 
r-s    s-t  _  (5-/)(/-r) 
x—t    x—r    {x—t){x—r) 


22. 


s-r    t-s      {s-t){t-r) 
x+t    r-\-x~'{x+t){x+r) 


24. 


ex    ,    dx         ,  J 
23.  — —zA — —^c+d 

{2X-sry^x-sr 
{2X-ssy    x-ss 

I      .     I         2X—p—q 
2  e    ■ 1 = i =— 

''■  x—p    x—q    x{x—p—q) 

26.  5.85(;+3-69=3.96  +  2.8:v 

27.  .374X— .53  +  1 .2:x;  +  .o6=.8  +  i  .32af 

28.  .3(1. 5^-. 8)  =  . 6(5.1  +  . 251;) 


232  First-  Year  Mathematics 

29.  .o5(20Jf — 3 . 2)  =  .8(4a;+  .  12)  — ii  . 256 

.2IiC+.OI2 

30.  1.4JC-1.61 =  1.3^ 

\-2X      2X-.S       2X-\      6.35-. 5^ 

31-  ~Tz Tnr^'        " 


32- 


•25         12.5  5  3 

.4x4-. 39     .2:)c— .66_  .o8:x;+ .38 


7  -9  -2 


Summary 


Algebraic  expressions  may  be  pictured,  or  graphed. 

An  equation  symbolizes  the  question: 

What  valine,  or  what  values,  if  any,  of  the  unknown,  are 
there  which,  substituted  for  the  unknown,  makes  the  first  mem- 
ber identical  with  tJie  second  member  ? 

The  identical  equation,  or  the  identity,  is  an  equation  that 
is  true  for  any,  or  all,  values  of  its  letters. 

The  distinguishing  sign  of  identity  is  =. 

The  conditional  e^iuation  is  an  equation  that  is  true  for 
one  value,  or  for  a  definite  number  of  values,  of  its  letters. 

The  left  side  of  an  equation  is  called  the  left  member,  or  the 
first  member,  and  the  right  side  the  right  member,  or  the 
second  member. 

An  algebraic  problem  is  stated 

I.  By  denoting  the  unknown  by  a  letter  and  then  translating 
the  verbal  statement  of  the  number  relations  into  a  symbolic 
statement  in  equation  form. 

II.  By  expressing  some  number  in  the  problem  in  two 
different  forms  and  writing  the  two  expressions  equal  to  one 
another. 

To  solve  an  equation  in  one  unknown  means  to  find  the 
number,  or  the  numbers,  which,  substituted  for  the  unknown 
(the  variable),  reduces  both  members  to  the  same  number. 


The  Simple  Eqiiation  in  One  Unknown  233 

Any  number  that  fulfills  this  condition  is  a  root  of  the 
equation. 

To  check  the  solution  of  an  equation  means  to  show  that 
the  result  is  the  root  of  the  equation. 

To  check  the  solution  of  a  problem  means  to  show  that 
the  result  answers  the  conditions  stated  in  the  problem. 


I 


CHAPTER  XI 

LINEAR  EQUATIONS  CONTAINING  TWO  OR  MORE  UNKNOWN 
NUMBERS.    GRAPHIC  SOLUTION  OF  EQUA- 
TIONS AND   PROBLEMS 

Indeterminate  Linear  Equations 
l6o.  The  meaning  of  the  solution  of  equations  and  of 
problems  is  made  clearer  by  means  of  graphic  methods. 

I.  How  many  8  and  lo  candle-power  bulbs  are  necessary 
to  obtain  an  82  candle-power  light?  (More  than  one  solu- 
tion.) 

Let  X  and  y  denote  the  number  of  8's  and  id's  respectively 

Then  Sx+  loy  =  82,  or  4»  +  5V  =  41 

41-4^ 
and  y  = . 


From  which,  if  x=  —  i 
then  y  =  0 


and  if  5C  =  14 
then,  y  =  —  3. 

These  solutions  are  the  co-ordinates  of  two  points  on  the 
graph;  viz.,  A  (-1,  9)  and  B  (14,  -3).     (Fig.  226.) 


^x  _::    _ 

Si 

ciffl           '>'iif 

-■     ^^E. 

% 

^„-..>-:i,c 

^T_r  M^ 

1            ^ 

t.              ^^f 

,  _E::fl  2:i.S._Stt^ 

^-7-c--f-h    ^^ ^ 

V 

^£ 

Gbl^ 

:      ~                  ^^ 

-       __    _                     ^i. 

Fig.  226 

Since  x  and  y  must  be  positive  integers  to  satisfy  the  prob- 
lem and  since  there  are  on  the  graph  only  two  points  whose 

234 


Linear  Equations  Containing  Unknown  Numbers      235 

co-ordinates  are  positive  integers,  namely,  C  (4,  5)  and  D  (9,  i), 
then  nc=4,  y  =  S\  and  x—g,  y=i  are  possible  solutions. 

What  are  the  co-ordinates  of  point  E?  Do  the  co-ordi- 
nates of  point  E  check  in  the  equation  ?    In  the  problem  ? 

What  good  reason  is  there  for  always  checking  in  the 
problem  ? 

What  is  the  origin?  What  are  the  co-ordinates  of  the 
origin  ? 

Find  the  co-ordinates  of  the  point  P.  (Find  L  P  by  pro- 
portion.) 

Where  in  respect  to  the  x-  and  ;y-axes  do  you  look  for 
points  having  positive  integral  co-ordinates  ? 

What  important  relation  is  there  between  the  equation  and 
the  co-ordinates  of  all  points  on  the  graph  of  the  equation  ? 

Why  may  8:x;-f-ioy=82  be  called  an  indeterminate  equa- 
tion ?    A  simple  equation  ?    A  linear  equation  ? 

Why  were  only  two  points  necessary  to  construct  the  graph  ? 

Why  should  the  two  points  which  determine  the  graph  be 
chosen  some  distance  apart  and  have  integral  co-ordinates  ? 

2.  A  contractor  has  cornice  stones  3  and  4  feet  in  length. 
How  many  of  each  may  he  use,  without  cutting,  to  lay  the 
cornice  of  a  wall  46  ft.  long?  Give  all  the  positive  integral 
solutions  of  the  equation. 

3.  In  how  many  ways  can  a  merchant  make  an  even  ex- 
change of  hats  at  $4.00  apiece  for  gloves  at  $3.00  a  pair? 
Give  the  least  positive  integral  solution  of  the  equation. 

4.  At  how  many  feet  from  the  fulcrum  will  a  75-pound 
weight  on  one  side  balance  a  45-pound  weight  on  the  other? 
Give  the  least  positive  integral  solution. 

5.  Weights  of  12  and  8  pounds  are  placed  at  opposite  ends 
of  a  lever  resting  on  a  fulcrum.  By  moving  the  1 2-lb.  weight 
3  ft.  toward  the  fulcrum,  the  weights  balance.     How  far  is 


236  First- Year  Mathematics 

the  fulcrum  from  each  end  of  the  lever  and  how  long  is  the 
lever  ?    (Several  solutions.) 

Let  X  and  y  denote  the  required  distances.  Then  x—^  will  denote 
the  number  of  feet  the  12-lb.  weight  is  from  the  fulcrum. 

Interpret  each  solution. 

What  is  the  least  positive  solution  of  the  equation  ? 

What  form  have  the  equations  of  problems  3  and  4  ? 

Which  term  is  missing  ? 

Their  graphs  have  what  point  in  common  ? 

What  are  the  co-ordinates  of  this  point  ? 

Do  the  graphs  of  all  the  equations  of  this  form,  ax  =  hy, 
pass  through  the  origin  ? 

Do  all  of  them  have  the  solution  x=o  and  v=o  ? 

Test  your  answers  with  the  following  equations:  4X  =  'jy, 
3^-f-23'=o,  and  4^— 5^=0. 

A  System  of  Two  Linear  Equations 

161.  Some  problems  lead  to  two  linear  equations  in  two 
unknowns. 

I.  Weights  of  12  lb.  and  8  lb.  are  placed  at  opposite  ends 
of  a  lever  resting  on  a  fulcrum. 

(a)  If  the  12-lb.  weight  is  moved  3  ft.  toward  the  fulcnun 
they  balance. 

(b)  Changing  the  weights  from  end  to  end,  and  moving  the 
12-lb.  weight  I J  ft.  toward  the  fulcrum,  they  balance  again. 

How  far  is  the  fulcrum  from  each  end  of  the  lever  ?  (See 
problem  5,  p.  235.) 

Letting  x  and  y  denote  the  required  distances,  then  from 

Condition  (a):  i2(.v— 3)=8y;  or  ^x—  2y=g  (i) 

Condition  (6):  8x=i2{y—i^),  and  2X  —  ^y=—4.         (2) 

Construct  the  graphs  of  (i)  and  (2)  on  the  same  axes. 
Find  the  co-ordinates  of  several  points  on  the  graph  of 
equation  (i).     Do  they  satisfy  equation  (i)  ?     Condition  (a)  ? 


Linear  Equations  Containing  Unknown  Numbers       237 

Consider  similarly  co-ordinates  of  points  on  the  graph  of 
equation  (2).  ^ 


-              -  z 

J 

_  _-         -       _        t  y 

unit                  2-.^ 

-        ~-          T/y^ 

a]/'- 

_   __          _        -^2p£i   -    - 

5^v 

A%t^ 

in      7.BKii_ 

^^'ESv 

_    _  ^^:i.  it^-    -0 

C^   H      7sa 

4^             -/ 

3>             -,2^ 

^                Z^ 

>^ 

/' 

7^ 

_i                         

Fig.  227 

What  must  be  true  of  the  co-ordinates  of  a  point  common 
to  both  graphs  ? 

What  is  evidently  the  solution  of  the  problem  ?  Check  in 
both  equations  and  in  the  problem. 

2.  Find  two  integral  numbers  such  that,  (a)  4  times  one 
diminished  by  3  times  the  other  is  equal  to  —22,  and  (h)  the 
sum  of  the  numbers  is  equal  to  —9. 

3.  Find  two  integral  numbers  such  that,  (a)  their  sum  is 
equal  to  —9,  and  {b)  6  times  the  first  diminished  by  5  times 
the  other  is  equal  to  45. 

4.  Find  two  integral  numbers  such  that  6  times  the  first 
diminished  by  5  times  the  other  is  equal  to  45,  and  the  sum 
of  3  times  the  first  and  2  times  the  second  is  equal  to  9. 

5.  Find  two  integral  numbers  such  that  the  sum  of  3  times 
the  first  and  2  times  the  other  is  equal  to  9,  and  4  times  the 
first  diminished  by  3  times  the  second  is  equal  to  —22. 

162.  It  was  seen  that  each  of  the  preceding  problems 


238 


First-Year  Mathematics 


produced  a  pair  of  linear  equations  having  one  and  only  one 
common  solution. 

Such  equations  are  commonly  called  a  system  of  two  linear 
equations. 

Exercise  XXXV 
163.  Solve  the  following  systems  by  the  graphic  method, 
and  check : 


(4) 
(5) 


2^-3y=4 
4^  +  5>'=3o 
8^  +  5>'=44 

2X—y  =  2 

(6)  ii:x;+7y=4o 
3^-5^=4. 


(i)       x-\-2y  =  i'] 
T,x-y=-2 

(2)  7-'^+3>'=-36 
Sx-2y=-s 

(3)  9*-6y=36 
13^:4-5^=11 

A  Pair  of  Contradictory  Linear  Equations 

164.  Find  two  numbers  such  that,  (a)  their  sum  is  10,  and 
(6)  3  times  the  first  increased  by  3  times  the  second  is  equal 
to  15. 

Letting  x  and  y  denote  the  required  numbers,  then  from 
Condition  (o)  x+  y  =  io 

Condition  (6)  3*+3>'  =  iS,  ora:+y  =  5 

That  is,  x-\-  y  =  io 

And,  at  the  same  time,  x-k-   y  =  5. 

Construct  the  graphs  of  the 
last  two  equations  on  the  same 
axes. 

Are  OB  and  OA  divided 
proportionally  by  D  C  ?    Why  ? 

Is  CD  parallel  to  AB? 
Why? 

What  does  this  tell  about 
the  problem?  The  equa- 
FiG.  228  tions  ? 


^ 

^                            vT.r 

\       _ 

^    ^    .f--. 

__    -^l^W       _     _    

^          ^^^ 

^;^     s_^^ 

'^   =^Sa 

s>^x  ^^ 

__     WiQ.'^^    ^■ii 

^^         ^ 

*S(^          S* 

''^             ^.^ 

-\             Rr-o 

^  \  a  ?M^l    \^H'■^'\ 

J                  •i\xtcx^s\ 

^               ^ 

^^    :  ^ 

Linear  Equations  Containing  Unknown  Numbers      239 

What  two  contradictory  statements  are  expressed  in  the 
equations  themselves  ? 

The  equations  of  such  a  pair  are  called  inconsistent,  or 
contradictory  equations. 

A  Pair  of  Equivalent  Linear  Equations 

165.  Find  two  numbers  such  that,  (a)  their  sum  is  8,  and 
(&)  ^  of  the  first  increased  by  J  of  the  other  is  equal  to  2§. 

Letting  x  and  y  denote  the  required  numbers,  then  from 
Condition  (a)       x+y  =  8 
Condition  (5)   ix+^y  =  2^,  or  x+y=8. 

The  graphs  evidently  coincide.  Hence  the  problem  and 
its  equations  have  an  unlimited  number  of  solutions.  Give 
several  of  these  solutions.  The  equations  of  such  a  pair  are 
called  equivalent,  or  dependent  equations. 

Exercise  XXXVI 

166.  Are  the  following  pairs  of  equations  contradictory, 
or  equivalent,  and  are  their  graphs  parallel,  or  coincident  ? 
Give  a  reason  for  each  answer. 

y  3.  yx-8   =4y-2X 

^•3^+r^  i8^-83'  =  i6 

4^+^=8  4.  3^-2>'  =  i4 

3  gx—6y=^6 

2.  x+^y  =  2  5.  2X-\-2y  —  'j—x  =  i2-^y 

X  ,  ,  2X+5y  =  2o 

-+\y  =  i 

2  6.  ^x+4y  =  i2 

6x+8y  =  i4. 

The  Solution  of  a  System  of  Two  Linear  Equations  by 
Algebraic  Methods 

167.  (a)  John  sets  out  on  a  walking  trip  and  travels  at  a 
uniform  rate  for  5  hours  when  he  meets  with  an  accident.  He 
continues,  however,  at  a  slower  pace,  and  3  hours  later  reaches 


240  First-Year  Mathematics 

a  point  26  miles  from  home,  (b)  If  he  had  turned  back  at 
the  time  of  the  accident  he  would  have  reached  in  3  hours  a 
point  14  miles  from  home.  What  was  his  rate  of  speed  both 
before  and  after  the  accident  ? 

I-«tting  X  and  y  denote  the  rate  before  and  after  the  accident  respec- 
tively, 

Condition  (a)       5:^  + 3^  =  26  (i) 

Condition  (b)       53^— 3^  =  14  (2) 

(Add.  Ax.)   ioa(;=4o  (Sub.  Ax.)  6y  =  i2 

(Div.  Ax.)        x=4  (Div.  Ax.)     y  =  2 

Check:  5(4) +3(2)  =  26,  and  5(4) -3(2)  =  14. 

Check  in  the  problem  also. 

The  equation  io:!c=4o  was  obtained  from  equations  (i) 
and  (2)  by  what  operation  ?  What  terms  were  canceled 
(eliminated)  from  equations  (i)  and  (2)  by  this  operation? 

What  is  true  of  the  coefficients  of  the  ;y-terms  which  makes 
possible  the  elimination  of  these  terms  by  addition  ? 

The  equation  6y  =  i2  was  obtained  from  equations  (i)  and 
(2)  by  what  operation?  What  terms  were  eliminated  from 
equations  (i)  and  (2)  by  this  operatibn  ?  What  is  true  of  the 
coefficients  of  the  j-terms  which  makes  possible  the  elimina- 
tion of  these  terms  by  subtraction  ? 

168.  What  must  be  done  when  the  coefficients  of  x  and  y 
are  not  alike,  as  in  the  following  example  ? 

Solve:  5:x;-l-3}'==26     (i) 


4x-yy  =  2       (2)  P  ' 
If  equation   (i)  is  multiplied  by  the  coefficient  of  x  in 
equation  (2),  and  equation  (2)  by  the  coefficient  of  x  in  equa- 
tion (i),  what  is  the  coefficient  of  x  in  each  of  the  resulting 
equations  ? 

What  operation  will  then  eliminate  the  af-terms  from  the 
resulting  equations  ? 

How  may  the  j-terms  be  eliminated  from  equations  (i) 
and  (2)  ? 


Linear  Equations  Containing  Unknown  Numbers      241 

Show  from  the  following  whether  your  answers  are  correct : 

7X(i),  3S»+2iy  =  i82)  .J,  4X(i),  2ax:-+-i2y  =  io4K  . 

3X(2),  i2x—2i.y  =  (i      5  SX(2),  205C— 35y  =  io    )^' 
^TX           =188  477  =  94 

X  =4  y  =  2. 

Having  found  the  value  of  x  as  shown  above,  the  value  of 

y  might  have  been  found  by  substituting  the  value  of  x  in 

one  of  the  given  equations,  thus 

5^+3^  =  26  Check  in  (i)  5  •  4  +  3  •  2  =  26 

S-4+3y  =  26 

3y=  6  Check  in  (2)  4  •  4—7  -2=   2. 

y=  3 

Do  the  values  x=^  and  y  =  2  satisfy  system  (a)  ?  System 
(h)  ?   System  (c)  ?   (a),  (6),  and  (c)  are  called  equivalent  systems. 

How  were  systems  (h)  and  (c)  obtained  from  system  (a)  ? 

From  your  answers  to  the  questions  above,  make  a  rule 
for  solving  a  system  of  two  linear  equations. 

169.  This  method  of  obtaining  from  two  equations  a 
single  equation  with  one  unknown  is  called  elimination  by 
addition  and  subtraction. 

Exercise  XXXVII 

I.  Tell  how  to  obtain  from  each  of  the  following  an  equiva- 
lent system  having  the  coefficients  of  the  first  unknown  the 
same;   of  the  second  unknown  the  same. 


(i)  4^+3y  =  i3 

(^) 

ix+sy='^i 

3^  +  2v  =  9 

ii^-3)'  =  5 

(2)  5;x;-|-4y  =  22 

(6) 

6(1—46  =  2 

3X_7),=  _i, 

5 

5a  +  7&=43 

(3)  3^-5>'  =  5i 

(7)   ■ 

-iix-j- 93^=16 

2X-\-']y=:^ 

^-\-2>y  =  2^ 

(4)     2/-7^=58 

(8) 

3«—    2V  =  ^ 

—9^  +  45=69 

-7M+i3v=-i 

(9) 

13^+ 

loy 

=  59 

lire— 

gy 

=  15- 

242  First-Year  Mathematics 

2.  Solve  the  systems  of  problem  i  and  check. 

170.  Solve 
2a-b    a  +  7 

3&-5  +  — — =-— ^  (i). 

4  5 

3  2 

Removing  2oX(i),     60  +  556  =  128  (3) 

denominators       6X(2),  34a  +  i56  =  i32.  (4) 

When  the  coefficients  of  an  unknown  have  a  common 
factor,  the  work  of  eliminating  that  unknown  can  be  made 
easier  by  multiplying  each  equation  by  the  remaining  factor 
of  the  coefficient  of  that  unknown  in  the  other  equation  after 
removing  the  common  factor. 

To  ehminate  the  a-tenns  from  equations  (3)  and  (4),  above, 
what  are  the  srftallest  integral  multipliers?  To  eliminate  the 
6-terms,  what  are  the  smallest  integral  multipliers  ? 

Show  whether  your  answers  are  correct  by  comparing  them 
with  the  following: 

i7X(3),  1020  +  9356  =  2,176  3X(3),     180+1656=384 

3X(4),  1020+  456=396  II X  (4),  3741  +  1656  =  1,452 

8906  =  1,780  3560  =  1,068 

6  =  2  <*=3- 

Supplement  the  preceding  rule,  §168,  so  as  to  include 
systems  like  the  foregoing. 

Exercise  XXXVIII 

171.  Tell  how  to  obtain  from  each  of  the  following  in  the 
first  column  an  equivalent  system  having  the  coefficients  of 
the  first  unknown  the  same;  of  the  second  unknown  the  same. 

1.  i23i;+i53;=66  3.  33.4—285=38 
i6x—25y=—2  22^+355  =  79 

2.  8^— 2i3'=33  4-  66m+55»  =  299 
^^+3S3'=i77  77W— i5n  =  2oi 


Linear  Equations  Containing  Unknown  Numbers       243 

5.  2i«+69z;  =  iii  7.  (2z*+3)  :  5  =  (3^+5)  :  7 

\^u  —  26v=2  'ju  :  4.V        =']']  :  40 

34<i-38ze;=-42  *     '^  4^5 

3  2 

9.  (2i2+3)(3r-7)  =  (3/2-5)(2r  +  i3)+29 
(5i?+7)(2r+9)=i2(ior  +  25)+iii 

10.  |M-§(w+i)=i(M-i)-9 
^(w+i)  — w  =  i(w— i)  —10 

x+y    x-y 

2  3 

3  4       * 

12.  a+i(3<i-&-i)=HI(*-i) 
iV(7&  +  24)=i(4a+3&). 

13.  Solve  each  of  the  12  systems  above. 


173.     Elimination  by  Substitution  and  by  Comparison 

Elimination  by  Substitution 

Solve:                                 5^—7^  =  1  (i) 

T,x+^y  =  i']  (2) 

From(i)     x=-^  (3) 

Substitute  (3) in  (2)   3^^^^"^^^"'^  ^"^^ 

(Mult.  Ax.)                3  +  2i3'  +  2o>'=85  (5) 

(Sub.  Ax.)                                41^=82  (6) 

(Div.  Ax.)                                      y=2  (7) 

Substituting  y  =  2  in  (3)               x—^.  .-   (8) 


244  First-Year  Mathemattcs 

Elimination  by  Comparison 
Solve:      $x  —  jy=i  (i) 

3:x;+4>'=i7  (2) 

From  (i)  x  =  ''-^^  (3) 

From  (2)  x=^^^—^  (4) 

3 

(Comp.  Ax.»)  ^=^^  <S) 

(Mult.  Ax.)               ^  +  2iy  =  8^  —  2oy  (6) 

(Add.  and  Sub.  Ax.)      41^=82        '  (7) 

(Div.  Ax.)                         y  =  2  (8) 

Substituting                       x=s.  (9) 

Exercise  XXXIX 

173.  Solve  each  of  the  following  by  the  method  of  substi- 
tution and  check  by  the  method  of  comparison: 

1.  4X  +  sy  =  i4  5-  9^-2r=44 
^x  —  2y=—i  6R—  ^=31 

2.  i^+fj=9  6.  i^u—6v=22 
^+iy=7  4u-\-gv=6i 

3.  'jx  —  2y=S  7.  7w— 22=46 
33f+4;y  =  i8  w-\-  2  =  13 

4.  2w+ii^=5o  8.  -|A+|^=33 
iiw4-2^=4i  ^h—^k  =  i'j. 

174.  If  the  time  required  to  do  a  piece  of  work  is: 

(i)  10  days,  what  part  of  it  is  done  in  i  day  ?  In  3  days  ? 
In  10  days? 

(2)  X  days,  what  part  of  it  is  done  in  i  day  ?  In  3  days  ? 
In  X  days  ? 

*  Comparison  Axiom:  Numbers  that  are  equal  to  the  same  nxmiber 
are  equal  numbers. 


Linear  Equations  Containing  Unknown  Numbers       245 

(3)  y  days,  what  part  of  it  is  done  in  i  day  ?  In  3  days  ? 
In  y  days  ? 

If  A  worked  3  days  on  a  piece  of  work  and  B  2  days,  they 
can  do  {^  of  it.  But  if  A  works  2  days  and  B  3  days,  they 
can  do  ^^^  of  it.  In  how  many  days  can  each  one  do  it,  working 
alone  ? 

Letting  x  and  y  denote  the  number  of  days  required  by  A  and  B, 

respectively,  then  -  and  -  will  denote  the  parts  A  and  B,  respectively, 
can  do  in  one  day. 

mence,    J+^=H  (i) 

X    y 

X     y 
These  equations  are  not  linear  in  x  and  y,  but  are  linear 

in  -  and  -  and  should  be  solved  for  -  and  - .     They  should 

X  y  X  y 

not  be  cleared  of  fractions.     From  the  values  of  -  and  - , 

X  y 

X  and  y  can  easily  be  found. 

What  operation  would  eliminate  the  >'-terms  if  their  numera- 
tors, were  the  same  ?    Solve  the  equations. 

Exercise  XL 

175.  Solve  the  following  without  clearing  of  fractions,  and 
check: 


X     y     ^^ 

3-  i^r^ 

II    2 

5-  iT-r* 

H=» 

13    ^i     1 

li=» 

^  ^^H 

-  li-* 

«■  M=H 

^^-* 

15    21 

^7=^ 

246  First-Year  Mathematics 

Systems  of  Linear  Equations  in  Two  Unknowns  Having 
Literal  Coefficients 


176.  Solve  for  x  and  y:  a'x-{-b'y=a-{-b 

(1) 

abx—aby=b—a 

(2) 

aX(i) 

a^x-{-ab'y=a'+ab 

(3) 

bX{2) 

ab''x—ab'y=b'—ab 

(4) 

(Add.  Ax.) 

(a3+ab')x=a'+b'' 

a'+b'        a^'+b'       i 

(S) 

a^-hab'    aia^'+b^)     a 


Find  the  value  of  y,   first  by  eliminating  the  A;-ternis  from 
ations  (i)  and  (2);  and  th( 
tion  (2).     Check  your  results. 


equations  (i)  and  (2);  and  then  by  substituting  x=-  in  equa 


Exercise  XLI 

I.  Tell  how  to  solve  each  of  the  following;    then  solve 
and  check: 


(i)      x-\-y=i 
ax—by=o 

(7) 

X    y    n 

(2)  cx+ny--=i 
ax—by=o 

I     II 

X    y  .  k 

(3)  ax+by=h 
bx-\-ay=k 

(8) 

ab 

X    y 

(4)  cx+dy  —  2cd 
bx—cy=d—c 

X    y 

(5)  ax+by=2ab 
2bx+^ay=2b''-\-^a' 

(6)  a^x-\-biy=Cj 
a2X-\-b3y=Ca 

(9) 

X     y 

n     m 
--I — =h 
X     y 

Linear  Equations  Containing  Unknown  Numbers       247 

,    .   5!; 4-1     h-\-i-\-k 


(10) 

2a  3& 
X      y 

X      y 

(II) 

x-\-a    a+c 
y-\-b    b^-d 

x-\-\     c+i 

y\\     d-\-x 

(12) 

X     a+b 
y     a  —  b 

x+c     a+b+c 

K'-dJ 

y+i 

h  +  i- 

-k 

x—y  = 

=  2k 

(14) 

x-b 
y—b 
x+b 
y+b 

b-c 
b+c 
36 -c 
3b+c 

(15) 

^   1   y 

a+b    a—b 

I 
a- 

-b 

a+b    a—b    a+b 


y—c    a— b-c 


Systems  of  Three  Linear  Equations 

177.  The  sum  of  3  times  the  first,  5  times  the  second,  and 
3  times  the  third  of  three  numbers  is  equal  to  22.  The  sum 
of  5  times  the  first,  and  3  times  the  second,  diminished  by  4 
times  the  third,  is  equal  to  —  i.  If  from  the  sum  of  4  times 
the  first  and  twice  the  second,  5  times  the  third  is  subtracted, 
the  remainder  is  —7.     What  are  the  numbers  ? 

Letting  x,  y,  and  z  denote  the  first,  second,  and  third  num- 
bers respectively,  the  equations  are: 

ZX+sy+T>z  =  22  (i) 

Sx+2,y-Az=-i  (2) 

^  +  2y-sz=-T.  (3) 

Eliminating  z  by  combining  (i)  and  (2),  and  again  by 
combining  (i)  and  (3),  or  (2)  and  (3),  what  two  unknowns 
will  the  two  resulting  equations  contain  ?  Can  these  equations 
be  solved  for  x  and  y  ? 

Having  found  x  and  y,  how  is  z  found  ?  If  we  eliminate 
first  z  and  then  y,  will  the  resulting  equations  contain  the 
same  two  unknowns?    What  two  unknowns  will  they  con- 


2^8  First-  Year  Mathematics 

tain?    Can  such  a  pair  of  equations  have  one  and  only  one 
solution  ? 

Eliminate  z  from  (i)  and  (2)  thus: 
4X(i)  i2X  +  2oy  +  i2Z=%%  (4) 

3X(2)  i5»+9>;-i22=-3  (5) 

(Add.  Ax.)  2']X  +  2()y^?,^  (6) 

Next,  eliminate  z  from  (i)  and  (3)  thus: 

5X(i)  i5af+25y+i5z  =  iio  (7) 

3X(3)  i2X^-ty-isz=-2i  (8) 

2']x-\-T,iy=^g  (9) 

Solving  (6)  and  (9),  :x;  =  i  and  y  =  2 
Substituting  in  (2),  5  •  1+3  •  2— 42=— i 

z=3. 
Check  in  (i): 

3 -1+5 -2+3 -3  =  22;   i.e.,  22  =  22. 

Check  in  (2) : 

5-i+3-2-4-3  =  -i;  i.e.,  -i  =  -i. 

Check  in  (3) : 

4-i+2-2-5-3  =  -7;  i.e.,  -']  =  -']. 

Write  out  a  rule  for  solving  a  system  of  three  linear  equa- 
tions. 

Compare  your  rule  with  the  following: 

Make  two  different  pairs  of  equations  out  of  the  three  equa- 
tions, and  eliminate  the  same  unknown  from  both  of  these  pairs. 
This  gives  two  equations  in  two  unknowns.  Solve  the  two 
resulting  equations  as  a  system  0}  two  linear  equations. 

Exercise  XLII 

178.  Tell  how  to  obtain  from  each  system  a  single  equa- 
tion containing  only  the  first  unknown;  the  second  unknown; 
the  third  unknown. 


Linear  Equations  Containing  Unknown  Numbers       249 


I. 

^x+y  —  2Z  =  —  i 

4.    a+36  +  5c  =  2i 

—4X  +  2y  +  T,z=g 

3a  — 2&  — 4C=22 

Sx+sy-2Z  =  s 

4a— 36— 6c  =  28 

2. 

2U  +  2V  +  W  =  g 

5.  5^  +  3>'=45 

U-\-T,V  +  2'W  =  l^ 

7;y— 22  =  27 

3M— 3z;+4W=9 

32  — 4;x;=  — 12 

3- 

i^x+^y  +  lz=^ 

6.     4M— z;  =  5o 

hx+h+hz=i 

5Z/— 2W=40 

i^+ij+iz=f 

6w— M  =  i5 

7- 

^x+4y  +  6z  =  2X-^6y  +  5z 

=  4a;  +  2_y+9Z=68 

8. 

X    y    z 

10.    7+5_2_33 

^--+-=U 

c     2     -? 
-  —  +-  =  5 

X    y    z 

X    y    z     "" 

X    y    z 

iC     ^     z 

9- 

V     u     ^ 

II.  ^—  =  2 

u    w 

:y4-2 

10       2 

rv  +  i     * 

12 

.  Solve  the  above  systems 

and  check. 

Problems  Involving  Two  Unknown  Numbers 

179.  The  student  should  make  a  practice  of  re-reading 
each  problem  until  the  conditions  imposed  upon  the  unknown 
numbers  are  fully  understood. 

I.  A  street  railway  company  receives  a  certain  sum  for 
each  cash  fare  and  a  different  sum  for  each  transfer.  On  one 
trip  the  conductor  of  a  car  collects  13  cash  fares  and  18  trans- 
fers amounting  to  $1.10  for  the  company.     On  the  return 


250  First-Year  Mathematics 

trip  there  were  7  cash  fares  and  24  transfers  and  the  amount 
for  the  company  was  $0.95.  What  does  the  company  receive 
for  a  cash  fare  ?    For  a  transfer  ? 

2.  Three  tons  of  hard  coal  and  two  tons  of  soft  coal  cost 
$32.  The  price  remaining  the  same,  2  tons  of  hard  coal  and 
6  tons  of  soft  coal  cost  $43 .  50.  What  were  the  prices  per  ton 
of  the  two  kinds  of  coal  ? 

3.  One  of  the  base  angles,  x,  of  an  isosceles  triangle  is 
equal  to  twice  the  vertex-angle,  y.  Find  all  the  angles  of  the 
triangle. 

4.  The  diflference  of  the  acute  angles  of  a  right  triangle 
is  36°.     Find  the  number  of  degrees  in  each  acute  angle. 

5.  The  diflference  of  the  adjacent  angles  of  a  parallelogram 
is  20°.     Find  the  values  of  all  the  angles  of  the  parallelogram. 

6.  The  consecutive  angles  a  and  6  of  a  parallelogram  are 
so  related  that  3a— 6=30°.  Find  the  values  of  both  of  the 
angles. 

7.  Three  times  one  of  two  adjacent  sides  of  a  parallelo- 
gram exceeds  twice  the  other  by  45,  and  the  perimeter  of  the 
parallelogram  is  80.     Find  the  length  of  the  sides. 

8.  One  dimension  of  a  rectangle  is  5  and  one  dimension 
of  another  is  3.  The  sum  of  the  areas  is  65  and  the  diflference 
35.     Find  the  dimensions  of  the  rectangles. 

9.  Two  trains  pass  each  other  going  in  the  same  direction, 
with  a  relative  speed  of  10  miles  an  hour.  Going  in  opposite 
directions,  they  would  pass  with  a  relative  speed  of  70  miles  an 
hour.     Find  the  speeds  of  the  trains. 

10.  The  sum  of  the  areas  of  two  rectangles  of  dimensions 
5  and  X,  and  3  and  y,  is  49,  and  the  area  of  a  rectangle  of 
dimensions  3  and  x  exceeds  the  area  of  the  rectangle  of  dimen- 
sions 5  and  y  by  9.     Find  the  dimensions  x  and  y. 


Linear  Equations  Containing  Unknown  Numbers       251 

11.  The  following  advertisement  was  published  by  an 
electric  company: 

The  price  of  electricity  has  dropped  since  1902  by  a  certain  num- 
ber of  cents. 

The  number  of  units  of  electricity  this  amount  of  money  will  buy  has 
increased  by  a  certain  number.  The  sum  of  these  two  numbers  is  59 
and  the  difference  is  19.     Find  the  nimibers. 

Solve  the  problem. 

12.  A  man  invests  part  of  $3,200  at  6  per  cent,  and  the 
rest  at  5  per  cent.  If  his  annual  income  is  $180,  how  much 
did  he  invest  at  each  rate  ? 

13.  A  man 'gained  8  per  cent,  on  one  investment  and  lost 
3  per  cent,  on  another.  If  the  money  invested  amounted  to 
$22,000  and  the  net  gain  was  $440,  what  was  the  amount  of 
each  investment  ? 

14.  Two  investments,  one  at  3^  per  cent,  and  the  other  at 
5i  per  cent.,  yield  annually  $150.  If  the  first  had  been  at 
8|  per  cent,  and  the  other  at  3^  per  cent.,  the  annual  income 
would  have  been  $175.  What  was  the  amount  of  each  invest- 
ment ? 

15.  The  areas  of  two  triangles,  having  equal  bases,  are  72 
sq.  in.  and  60  sq.  in.  Twice  the  altitude  of  the  first  plus  3 
times  the  altitude  of  the  second  is  equal  to  54  inches.  Find 
the  altitudes. 

r6.  The  areas  of  two  circles  are  to  each  other  as  4  is  to 
36.  One-half  the  radius  of  the  first,  plus  \  of  the  radius  of 
the  second,  is  equal  to  6\.  Find  the  radii  of  the  circles  and 
their  areas. 

The  formula  for  the  area  of  a  circle  is:  A  =Tr».  Find  the  ratio  of 
the  radii.     . 

17.  The  ratio  of  the  circumferences  of  two  circles  is  2, 
and  6  times  the  radius  of  the  first  minus  4  times  the  radius 


252  First-  Year  Mathematics 

of  the  second  is  equal  to  14.     Find  the  radii  of  the  circles  and 
their  circumferences. 

The  formula  for  the  circumference  of  a  circle  is:  c  =  2Jrr. 

1 8.  The  altitude  of  a  trapezoid  is  8  and  the  area  is  56. 
If  the  lower  base  is  increased  by  a  length  equal  to  the  upper 
base,  the  area  is  72.     Find  the  bases  of  the  trapezoid. 

19.  The  lower  base  of  a  trapezoid  is  24  and  the  area  is 
150.  If  a  length  equal  to  \  of  the  lower  base  is  added  to  the 
upper  base  the  area  is  170.  Find  the  altitude  and  upper  base 
of  the  trapezoid. 


Exercise  XLIII 
20.  Solve  fhe  following: 
(I) 


(2) 
(3) 


s-\-(ip=^s  +  $p 
{s+A)-{sp-s)-s=o 

iR-\r=S 
^R-ir  =  7 

(  mx—ky=mR 

\  kx-\-my=kR  (for  x  and  y) 

(4)  ]  25 

(Ki2+ir)-i(ie-4)=4 


-=— 12 


(5) 


(6) 


R    r     ' 

R^r    ^ 

20  — 2K     5K    H    aK  —  iq 


Linear  Equations  Containing  Unknown  Numbers       253 

i  2/>  +  25  +  3Z/  =  4 

(7)  ]  3/'+4^+6z;  =  7 
(   p-\-2s-\-6v=^ 

(8)  ]  K+R  =  2a  (for  H,  K,  and  R) 
{R+H  =  2h 

(9)  Construct  the  graphs  of  (2)  and  (6) . 

21.  In  the  right  triangle  A  B  C,  C  D  is  perpendicular  to  the 
hypotenuse  AB.     Find  m  and  n. 

The  segm2nts  of  the  hypotenuse  are  to  each  other  as  the  squares 
of  the  two  sides  adjacent  to  the  segments. 


Fig.  229 


Fig.  230 


22.  In  the  right  triangle  E  F  G,  /t  is  perpendicular  to  the 
hypotenuse  E  F.     Find  m  and  h. 

Apply  the  principles  of  similar  triangles. 

23.  In  Fig.  231  51;  has  the  same  value  throughout.     The 
same  is  true  of  y.     Find  x  and  y  from  the  relations  indicated 


Fig.  231 

(i)  in  triangles  i  and  2;  (2)  in  triangles  i  and  3;  (3)  in  triangles 
2  and  3. 


254 


First-Year  Mathematics 


24.  The  perimeter  of  the  first  triangle,  Fig.  232,  is   288. 
Find  X  and  y  from  the  relations  indicated  in  the  triangles. 

25.  The  corresponding  altitudes  of  two  similar  triangles 
are   represented   by   the   expressions    T,x-\-/^y—'j   and    2X—y 


3x-2y^90 


Fig.  232 


respectively,  and  the  corresponding  bases  by  the  expressions 
^x—2y+i  and  4X  —  2y  respectively.  The  perimeter  of  the 
first  triangle  is  4  and  of  the  second  is  6.  Find  the  values 
of  X  and  y. 

The  perimeters  of  two  similar  triangles  are  to  each  other  as  the 
corresponding  sides  or  altitudes. 

26.  In  Fig.  233  D  E  is  parallel  to  B  C.  The  sides  A  B  and 
AC  are  represented  by  the  expressions  4X-\-^y+i   and   6x 

+4J  +  2    respectively.      Find 
X  and  y. 

27.  A  number  of  two 
digits  is  equal  to  9  less  than 
7  times  the  sum  of  the  digits. 
If  the  digits  are  reversed  the 
number  is  18  less  than  the 
original  number.     What  is  the  original  number? 

28.  Three  times  the  reciprocal  of  the  first  of  two  numbers 
and  4  times  the  reciprocal  of  the  second  are  together  equal 
to  5.  Seven  times  the  reciprocal  of  the  first  less  6  times 
the  reciprocal  of  the  second  is  equal  to  4.  What  are  the 
numbers  ? 


Linear  Equations  Containing  Unknown  Numbers      255 

29.  If  the  numerator  of  a  certain  fraction  is  increased  by 
2a  and  the  denominator  by  3&,  the  resulting  fraction  is  equal 

to  r.     If  the  numerator  and  denominator  are  decreased  by 

2a 
1 2a  and  66  respectively,  the  resulting  fra,ction  is  equal  to  -r  • 

What  is  the  original  fraction  ? 

30.  A  and  B,  working  together,  build  a  fence  in  4  days. 
They  can  also  build  it  if  A  works  3  days  and  B  6  days.  In 
how  many  days  can  each  alone  build  the  fence  ? 

31.  A  boat  crew  rows  4  miles  down  stream  in  20  minutes 
and  the  same  distance  up  stream  in  35  minutes.  Find  the 
rate  of  the  boat  crew  in  still  water  and  the  rate  of  the 
current. 

Exercise  XLIV 

180.  Solve  the  following  exercises: 

-Ss+gp=4-  lr-2jv+^ 

'js+6p=—8o  )^—3    ^  +  5 

5R  +  i4r=3&5  ^')r_±l^v±s 


5R-\-9r=So  ^^+^    ^-4 

w— 2a+4c_3& 
^  — 2a+36     4c 
'w-\-4c_2a+^b 

-+-=^  \         k  +  3b~2a+4C 

^     4  (for  w  and  k) 

83     ^ 


— *o  =  io 
32 

m  .  n 


ah—ak  ,  (b—c)h    2(c—b)k    ,        ,r     1       ,  i.\ 

h ^^ i-=zb—c  (for  h  and  k) 

—4a  4a  2a 

(a—b-\-c)k  =  {a+b—c)k 


2c6  First- Year  Mathematics 


ah  I      w    V    2r 

70    12  ^^    )      5,8,  6 

i =12  lO-  (      — I 1 4 

a      b  \     w    V    2v 

^  1  12  ,  18    10 

1 =1  \    V        2V       W 

245 

A     B    C_  ( 6Cx+aC2=46+a 

2      2     3  ~~  "  I  aCi+6C2=4a— 6 

ABC  (fo^^  ^i  ^°^  ^«) 

=  -13 

442 


12. 


13- 


14. 


15- 


{m+n)R^  +  {m-n)R^  =  2{m'+n'')      ^ 
17  — 2C    5(^—8 


o  —  ^j- 

3             2 

16  — 2d 

2c-rf     S5+C 
4            5 

2.2g  + 

.8     .sh-.4 

1-5 

•3 

•5^- 

•3     i-5^--5 

.2 

2-5 

6-4i? 

3H-8 

3 

2 

3^-4  = 

8i?-2 

= 1 

5 

Given  5  =  Ja/,  and  T  =  Ra.     (i)  Solve  for  /  and  a;  (2) 
Solve  for  R  and  a. 

,5   /  (3)  When  in  the  equations  R  =  2,  t=6,  and  5=81, 
what  do  T  and  a  equal  ? 
(4)  Check  by  substituting  in  the  fornjiula  obtained  by 
solving  (i). 


Linear  Equations  Containing  Unknown  Numbers       257 


17- 


Given/=a  +  (»  — 1)</ and  5=-(a4-0-     (i)  Solve  for  a 

and  /;  (2)  Solve  for  a  and  d 
Find  the  values  of  the  remaining  letters 

when  (i),  w=8,    d=2,      and  ^  =  104 

when  (2),  n  =  i6,  a  =  5,      and  5=40 

when  (3),  a  =  7,     /=49,     and  5=812. 

Given  l=ar"~^ 

a(r~-i) 

and  5  =  -^^ 

r—i 

(i)  Find  a  and  /  when  5  =  765,    r  =  2,     and  »=8 
(2)  Show  that /r=ar» 
s—a 


(3)  Show  that  r- 


-I 


three  unknowns. 


(4)  Find  r  and  w  when  a =3,    ^  =  384,     and  5  =  765. 

Problems  Involving  Three  Unknown  Numbers 
181.  Some  problems  lead  to  systems  of  three  equations  in 

8 

A 


1 .  In  the  triangles  ABC 
and  k'WC,  B  =B',  A  B  =  A'B', 
and  B  C=B'C',  and  the  sides 
may  be  designated  as  shown 
in  Fig.  234.  Find  the  values 
of  X,  y,  and  z. 

2.  The  triangles  of  Fig. 
235  are  equal  in  all  respects. 


2S8 


First-Year  Mathematics 


Find  the  values  of  x,  y,  and  z,  the  sides  being  designated  as 
folbws: 

!di=x-\-T,y—z,    and  a'  =  io 
h  =  2X+^y-\-^,    and  b'  =  i7 
c  =  T)X—y-\-2Zy    and  c'  =  i5 

!a  =  2}'+30,     and  2i'  =  i/^—x 
h  =  2{x-\-z),    axidh'  —  io—y 
c=3:x;+4)',     and  0^=32+2 

!=2(  2X-\-y),     and  a'  =  i3— z 
b=x+z,     andb'=>'+4 
c=je  +  2)',     and  c'=z 

!a  =  2:!C+3Z,     and  a'  =  2(4;y  — 2J) 
b=a£;+5z,     and  b'= 4^+8 
c=3:»— 2,    and  c'  =  5(2>'— 4). 


3.  The  triangles  of  Fig.  236  are  similar  and  the  ratio  of 
their  perimeters  is  3.  Find  the  values  of  x,  y,  and  z  under  the 
following  conditions: 


Linear  Equations  Containing  Unknown  Numbers       259 

!a.  =  2X-\-y-\-2Z,     and  a' =  7 
b=x+4y—z,    and  b'  =  5 
c=—x+^y+5z,    and  c'=6 


a  =  75(;+z, 


(2)   (h=x-{-z, 


C  =  'JZ-\-2X, 


and  a'=io+- 
3 

and  b'=4+^ 
3 

and  c'==5+- . 
3 


4.  The  sum  of  the  three  digits  of  a  number  is  1 6.  If  the  order 
of  the  digits  is  reversed,  the  new  number  is  396  less  than  the 
original  number.  If  the  middle  digit  be  placed  first,  the 
resulting  number  is  90  less  than  the  original  number.  What 
is  the  number  ? 

5.  A  and  B  can  do  a  piece  of  work  in  35  days.  B  and  C 
can  do  it  in  17J  days.  C  and  A  can  do  it  in  21  days.  How 
long  will  it  take  each  to  do  it  alone  ? 

6.  In  the  triangle  ABC,  Fig,  237,  find  the  values  of  x,  y, 

and  2. 

(i)  Ifa=6andb=8,      c=8:»  — 7;y 

(2)  If  a  =  5  and  b  =  i2,    c  =  $y-{-;^z 

(3)  If  a  =  9  and  b  =  i2,    c=4x+;^z. 

7.  The  angles  of  a  triangle  are  A,  B,  and  C;  \A-{-^B=C, 
a,ndiA+Y^B  =  i^C+^o.   Find 
the  values  of  yl,  B,  and  C. 

8.  $18,000  is  invested  as 
follows:  One  part  at  3^  per 
cent.,  a  second  part  at  5  per 
cent.,  and  the  rest  at  4  per 
cent.,  and  the  total  annual  in- 
terest is  $730.  If  the  first  part  had  been  invested  at  4  per 
cent.,  the  second  at  3  per  cent.,  and  the  third  at  6  per  cent.. 


26o  First- Year  Mathematics 

the  total  annual  interest  would  have  been  $840.    How  much 
was  each  part  ? 

Summary 

Problems  involving  two  unknown  numbers  may  be  solved 
graphically. 

The  graph  of  a  linear  equation  is  a  straight  line. 

The  solutions  of  a  linear  equation  are  the  coordinates  of 
points  on  the  graph  of  the  equation. 

The  coordinates  of  two  points  are  sufficient  to  locate  the 
graph. 

The  coordinates  of  every  point  on  the  graph  of  an  equation 
are  solutions  of  the  equation. 

Positive  integral  solutions  and  least  positive  integral  solu- 
tions of  linear  equations  may  be  found  from  the  graph  of  the 
equation. 

The  solution  of  a  problem  involving  two  linear  equations 
is  shown  to  be  the  coordinates  of  the  point  of  intersection  of 
the  graphs  of  the  equations. 

The  graphs  of  a  pair  of  dependent  equations  are  two  parallel 
lines. 

The  conditions  of  a  problem  having  an  infinite  number  of 
solutions  are  represented  graphically  by  two  coincident  lines, 
the  graph  of  two  equivalent  equations. 

Two  equations  having  all  solutions  in  common  are  called 
equivalent  equations.     Their  graphs  are  coincident  lines. 

The  method  of  elimination  by  addition  and  subtraction  ex- 
plained. 

The  method  of  elimination  by  substitution  explained. 

The  method  of  elimination  by  comparison  explained. 

The  solution  of  literal  equations  and  equations  involving 
the  reciprocals  of  the  unknown  numbers  explained  and 
applied. 

A  system  of  two  equations,  that  are  linear  in  the  reciprocals 


First-Year  Mathematics  261 

of  the  unknowns,  of  the  type:  — \—=-,,  may  be  solved  as 

X    y    a 

linear  equations  by  regarding  -  and  -  as  the  unknowns. 

oc         y 

The  solution  of  systems  involving  three  unknown  numbers 

explained  and  applied. 

Problems  involving  two  and  three  unknown  numbers. 


CHAPTER  XII 
FRACTIONS 

182.  The  operations  of  addition,  subtraction,  multiplica- 
tion, and  division  are  extended  to  apply  to  fractions. 

Reduction  of  Fractions 

1.  How  many  fourths  equal  |?  How  many  sixths?  How 
many  eighths  ? 

2.  How  many  halves  equal  ^^P  How  many  fourths? 
How  many  sixths  ? 

3.  How  many  sixths  equal  J?  How  many  ninths?  How 
many  twelfths  ? 

4.  How  many  fourths  equal  ^|?  How  many  eighths? 
How  many  twelfths  ? 

5.  Show  that  ^=H=i 

d  I 

6.  Show  that  —  is  equal  to  - . 

ac  c 

01  1       ^  •  ,       dh 

7.  Show  that  -  IS  equal  to  —  . 

c  ac 

8.  Make  a  rule  for  reducing  fractions  to  lower  terms;  to 
higher  terms. 

183.  Multiplying  numerator  and  denominator  of  a  frac- 
tion by  the  same  number  does  not  alter  the  value  of  the 
fraction.  By  this  principle  we  may  reduce  a  fraction  to  higher 
terms. 

184.  Dividing  numerator  and  denominator  of  a  frac- 
ion  by  the  same  number  does  not  alter  the  value  of  the 
frciction.  By  this  principle  we  may  reduce  a  fraction  to 
lower  terms. 

262 


Fractions  263 

Exercise  XLV 

185.  Reduce  the  following  fractions  to  lowest  terms,  doing 
as  many  as  you  can  mentally: 


10.  —, —  16.  ,;    ,  7 

12  aky^  b(x-^yy 

00  a'^h^m  rri^ix—yY 


a^b^'m"  '     ms(x—y) 


u 

T5T  ■  u'v^y''  '  rx-\-ry 

183"  o(^  amn—bmn 


u^v^y^  -    cx-\-cy 

12.  —r-T^^  18. 


13-  -:;rT;  ^9 


i^f^  "^  r)c;"+3  ^    am3n'-bm3n' 

ab  3C"+'  (a+&)'(&-c)3 

abc  3f"+3y»  +  S                                  (^^_^)3(w_w)7 
I"?. 21.    -^^ ^—^ — 

acy  -;pn+ii^n+i  {u+v)^(m—ny 

a3hii(r-s)9  a3bsc''9(a+by(a'+b''y 

^^'  a%^5(r-s)5  ^^'  a'b''c93(a-{-by{a'+b')s' 

186.  Adding  fractions  that  have  the  same  denominator. 


a    h 
c    c 

m    n    k 

XXX 

r_5     /    V 
y    y    y    y 

3  .^    5 


Exercise  XL VI 

the  following 

fractions 

mentally: 

I. 

f+f 

8 

2. 

i+f 

3- 

i+t-f 

4- 

tV+A- 

A+tV 

9- 

5- 

^-l-f+f 

6. 

l-l-l- 

-¥ 

10, 

7. 

a    6 

II. 

a+b    a+b    a+b 

12.  Make  a  rule  for  adding  fractions  having  the  same 
denominator. 


ion: 
I. 

H+i^+H 

2. 

X    y 

2      2 

3- 

H+A-H 

4- 

X    y 

"T" 

a    a 

5- 

a    h     c 

XXX 

6. 

a-\-h    a—h 

2               2 

264  First-Year  Mathematics 

Exercise  XLVII 
Combine  each  of  the  following  expressions  into  a  single 


x—y  ,  2y 

7.  --\-— 

a        a 

c—ax    c  +  ^ax 

o. 1 

45  45 

3a     T,a—Sb 

10.  — + ^  - 

17a    17a     17a 

Sa  —  b     2a  — 3& 
x—y       x—y 

187.  Adding  fractions  that  have  different  denominators. 

1.  Reduce  §  and  ^  to  fifteenths. 

2.  Reduce  -|  and  f  to  fractions  having  the  same  denomi- 
nator.    (See  §183.) 

3.  Reduce  —  and  -  to  fractions  having  the  same  denomi- 
nator.    (See  §183.) 

4.  What  is  the  sum  of  \  and  ^?     \  and  -^?     \  and  ^? 
T^and^ij? 

5.  What  is  the  sum  of  *  and  4?   i+i  =  -^  +  -^=^-i^  . 

''^^2-72-77-2 

Give  reasons. 

6.  In  the  same  way  give  the  simi  of  i  and  \.    ^  and  f . 
-h  and  ^^^ 

Fractions  having  i  for  numerator  are   called    unit  frac- 
tions. 

7.  Make  a  rule  for  writing  at  once  the  sum  of  two  unit 
fractions. 


Fractions  265 

Exercise  XL VIII 
188.  Using  the  rule  find  these  sums  mentally: 

'•  III  «•  -+- 

2.  1+^  n    m 

3-  i+i  I       I 

4.  \-\-\  a+b    a 

6-  i+i  ".  Z7^+:    ' 


2a+36     3a— 26 

1,1  1,1 

7-  -+7.  12.  — r-— ;+: 


X    y  2X^  —  ^y'     x^+4y' 

13.  What  is  the  difference  of  §  and  J  ?    J  and  ^  ?    ^  and  4^? 
^  and  ^  ?    i  and  ^  ? 

14.  Write  out  the  work  of  the  parts  of  problem  13  as  is 

7         ■?       7  —  ■> 
done  in  this  explanation.     4—1=-^ ^  =^ — ^ .     Give  rea- 

^  ^     '     5-7     5-7     5-7 

sons. 

15.  Make  a  rule  for  writing  at  once  the  difference  of  two 
unit  fractions. 

Exercise  XLIX 
189.  Apply  the  rule  to  find  these  differences: 

2.  \-\  ^'  h    X 

1     I  ft    ^     ^ 

3-  i--  6.  --^ 


4-  i-^  7.  7- 


t     m 


266  First-Year  Mathematics 


8.  l--^ 


b     a+b  x'+y'    x'—y' 

I  I  _2 I 

a+b    a—b  '  a+b    a+b+c 

I  I  ->.        5 


ia+^b    ^a  +  2b         '        a+b 
hat  the  addition  and 
may  be  indicated  thus:     i±iV  = 


14.  Show  that  the  addition  and  subtraction  of  unit  fractions 

ii±5 


5-II 
190.     Addition  and  Subtraction  of  Fractions 

:.Addt.„di.    in-S^S^U^'-^- 

2.  Subtract  ^  from  f . 

ft  ft 

3.  Add  -J-  and  ^  where  «i  (read  "«  one")  and  i,  (read 

"<i  one")  stand  for  first  numerator  and  first  denominator; 
«a  {n  two)  and  Jj  {d  two)  stand  for  second  nimierator  and 
second  denominator. 

di     dj     didj     d^di  d^d^       ' 

5.  Subtract  ^  from  -^  . 

d,  d, 

6.  From  3  and  4  obtain  a  rule  for  writing  at  once  the  simi 
or  difference  of  two  fractions. 

7.  By  this  rule  write  the  siun  and  difference  of  7  and  -  . 

0         a 


Exercise  L 

By  this  rule  give  the  values  of  the  following  indicated  sums 
and  differences,  giving  as  many  as  you  can  mentally: 


Fractions  367 

I-  *+A  ^   x.y  „    a+3    a+5 

2.  f-f  '    a    a  5  7 

3-  i^u+T^r  08.7  a+x    a—x 

8.  -±-  12. — 

_  I  7  ^    y  a—x    a-\-x 


3 


:»  ,  c  2X  —  'za     2X—a 

±1  13. 


-   <*  I    7  yd                      x—2a      x—a 

y  a?    ic                      ;x;^  +  2a^     x'  —  2a' 

5   ^_j_^  ^°'  a    b  ^^'   x^'+a'  ~  x'-a'  ' 
'  5     10 

15.  AddT^^+f. 

5-8+3. i2_  76 
T^+*-      8.12      -8.12-**' 

This  method  is  especially  useful  in  adding  two  fractions  whose 
denominators  have  no  common  factors. 

In  case  the  denominators  of  the  fractions  to  be  added  have 
a  common  factor,  the  fractions  should  be  reduced  to  fractions 
having  for  their  denominators  the  least  common  multiple  of 
the  given  denominators,  and  then  added,  thus: 

16.  This  method  is  used  in  adding  three  or  more  fractions, 
thus: 

17.  Add  ^+---. 

X'     xy    y' 

ah      c  ■__  ay'      hxy      ex'  _ay'-\-bxy—cx' 
X'    xy    y'~x'y'    x'y'    x'y'~         x'y' 

Exercise  LI 
191.  Simplify  the  following: 

,.  P+i+JL  ,.  ^+Sc^7l 

2     5     10  362 


268  First-Year  Mathematics 


%.  - — ^-  17.  ii:x;— ^— 

•^    12     20  '               3 

a    ax  4 

ah  ah 

c.  — I —  19. 1 

''    ex    cy  x'y    xy' 

.lb  a-i-b  ,  a—b 

o. 20. 1 

c    ca  X         ^x 

X       z  a— 2b    4a  — 56 

i2ah    6b  '     2)^           $x 


a—b    h—c  . c—a 


I       I.I 


8.  -i-+-r-+ 22.  ---  +  ^ 


2      ly^  ^  xy^Ary"  ^     A     ^ 

xy    xyi       x^y^  23.  ^+-+^j 


a  a& 

ro. 


I        I 


a-i     a(a-i)  24.  ^__^    ^ 

II.  -^ 7-^-^  ,    53C-4y+3Z  .  23c+3y-4Z 

iC  — I       2[x  —  \)  25.    =^^ =^ —-\ 

'"•  ;^+4^^=^  26    7^+3^-4^,2^+4^-3^ 

XT  ah 
Note. — 

27 — — 

b              ab  '■  x^y    x'^y''    xy' 

'^-  ^2b    sad  +  6bd  ^^j       ^ 

14.  -+b  ^    ^-^ 

6  20.   (^  +  ^)'     I 

Note. — Put&=-.  '^'       4a6 

^  a;      ,  jc+3    x-s 

16.  sa+^  31.    ^^  -   "^         '''' 


3  c+<i     c— </     c^ 


Fractions  269 

192.     Multiplication  of  Fractions 

1.  Multiply  I  by  8 ;  by  1 2 ;  by  5;  by  25;  by  a;  hy  xy. 

2.  Multiply  J  by  i;  ^by^;  §byi;  ^byj;  iby|;  ^by^ 
(i  oi  J,  iXj,  and  ^  •  J  are  all  equivalent). 

3.  Multiply  I  by  4;  by  f ;  by  f ;  by  ^;  by  ^;  by-;  by  ^  . 

530 

4.  Make  a  rule  for  multiplying  two  fractions  together. 
Fractions  are  multiplied  hy  tmdtiplying  their  numerators  for 

the  numerator  of  the  product,  and  multiplying  their  denomina- 
tors for  the  denominator  of  the  product. 

5.  Multiply  H  .  if. 

Since  12  •  15  will  be  the  numerator  of  the  product  (why?), 
and  35  •  16  will  be  the  denominator  of  the  product  (why?), 
any  factor  of  12  or  15  which  is  also  a  factor  of  35  or  16  will 
divide  both  the  numerator  and  the  denominator  of  the  prod- 
uct. It  is  simpler  to  divide  out,  or  cancel,  such  factors  in 
advance,  thus: 

3     3 

>«'  ><r  3  3  9     ^. 

X  ■  )js"i '  4^28  ■     ^^'^^  reasons^ 
7     4 


Exercise  LII 
193.  SimpUfy  the  following: 

2.  tXf  0    d  X    a 

3-  f  Xf 

4.  f XJ  8.  ^X^  II.  ^X^' 

yd  di     da 


5.  ixi 

-X- 

5     7  "■  i^\  "■  b/^b. 


6.  ^X^-  0.  ?X^  12.  Px?^. 


270  First-Year  Mathematics 


EXERCTSE   LIII 

194.  Find  the  values  of  these  products,  using  cancellation 

when  possible: 

a    c    X                  „   a'    b'     b 
^'  b     d    y                    '  b^     a^     a 

3-    t       7       J 

dj     dj     d^              ^'  a'     6     a 

a     a'    a3                  a    x^    a' 
^"  6»  '  y  *  fes            ^°'  X  '  a' '  x"^ ' 

9.  Make  a  rule  for  multiplying  any  three  fractions  together. 

10.  Change  your  rule  to  make  it  apply  to  the  product  of 
4  fractions;   of  n  fractions. 

11.  Make  a  rule  for  dividing  out  or  cancelling  factors  in 
the  multiplication  of  any  number  of  fractions. 

Exercise  LIV 
195.  Perform  the  indicated  operations,  using  cancellation 
when  possible: 


2.    *  •  T^^ 


9-   TXT, 


I 

3.  -  •  oc  10. 

I 

4.  —  •  a3  II. 
a' 


a  I 
b'x 
a  c 
c    d 


12. 


13- 


7.  -^  •  gabc  14. 

8    ~  .^ 
'  xy'  be  ^^'  &y+8x        3 


2ab 
2>xy 

^ax 
6by 

I  safe 

24xyz 

i6xy 

2sbc 

Sab^ 
4xy' 

56c     JXZ 
6yz    Sac 

2a'x 
Sb-y 

6by'     56 
yax'     4a 

a^b^ 

tx^y'^ 

4X^y3 

Sa^b' 

a+b 

a'-b^ 

a—b 

a^'+b'' 

2'JX 

x-\-y 

1 6. 


Fractions  271 

a  b  a  +  i     a  +  2     a-{-;i 

a+b    a—b  "'  a  —  i     a— 2     a— 3 


6{x—y)    -i.$x^y2  x^—xy     (a+6)* 

1 7,  • 20.  •  

Sxy^      ^{x-y)  za  +  Tfi     {x-yY 

a'—ab     x^+xy 


18. 


x^—xy     a^+ab 

21.  Show  that— Y=H — —  =  -\ -. 

0  0  —0 


Division  of  Fractions 
Exercise  LV 
196.  Multiply  the  following  fractions: 


I. 

2. 

1 

6. 

y     X 

10. 

a     I 
I     a 

3- 

4. 

x 

it 

7 

3 

X 

7. 
8. 

n    d 
d     n 

II. 
12. 

x^      y 
y     xs 

a+b    c+d 

5- 

X 

10 

9- 

¥-tV 

c+d    a+b 

10 

X 

13.  What  is  the  simplest  form  of  the  product  in  each  of 
the  problems  1-12  ? 

14.  By  examining  the  products  of  problems  i-i  2,  tell  without 

actually  dividing,  how  many  times  f  is  contained  in  i ;  f  in  i ; 

f  in  i;    f  in  i;    ^  in  i;    |^  in  i;  2  in  i;   12  in  i;  a  in  1; 

b  . 

-mi. 

a 

15.  Give  a  quick  way  of  finding  how  many  times  any  frac- 
tion, or  any  whole  number  is  contained  in  i. 

16.  How  many  times  are  these  numbers  contained  in.  i  ? 

(I)  i  (2)  I  (3)  I 


272  First- Year  Mathematics 

(5)i  W?  (9)^ 

17.  A  certain  number  is  contained  in  i,  ^  times;  how 
many  times  is  it  contained  in  2  ?   in  3  ?   in  5  ?  in  8  ?  in  20  ? 

in  a  ?  in  6  ?  in  f  ?  in  f  ?  in  ^^^  ?  in  r  ?  in  -^  ?  in  -  ? 

18.  How  many  times  is  f  contained  in  the  following  num- 
bers? 

(i)int?  (4)miV?  (7)  in  a? 

(2)in^?  (5)inT^?  (8)  in  |? 

(3)  in  I?  (6)  in  12?  (9)  in  -  ? 

y 

19.  Find  the  result  in  the  following  indicated  divisions: 
(I)  f-f  (3)  ^-f  (5)  ^-^ 

(2)4-i^r  (4)  t-i  (^>^-^7- 

One  fraction  is  divided  by  another  by  multiplying  the  divi- 
dend by  the  inverted  divisor;  that  is,  by  multiplying  the  divi- 
dend by  the  reciprocal  of  the  divisor. 

Why  is  the  divisor  inverted  ? 

Why  is  the  inverted  divisor  multiplied  by  the  dividend  ? 

Exercise  LVI 
197.  Perform  the  indicated  operations: 

y  i6:x;* 

.     xy                                                        loa" 
2.  ab-. — T  4.  2^a^-. 


Fractions  273 

^  -^  ah  ^        '     ^ab 

,        8X3^323 

7.   (32:x:3j2z^— 40:x;='j322)h ^ 


27:x;s^3     ()x^y*  X"'^^     :x;"+3 

2o:x;*3'3  ^    4a:x;3'  a*~^     a* +3 

2ia4cs  ■  ^a'b'c'  '  6'"+^  '  b"*-' 

^oa^b^c^  ^  S5a%^c^ 
22m^x'^z^  '  ?)Sm^xz'' 

2^{x  —  iy  2>^{x  —  \) 


13- 
14. 


7o(a=-6^)  ■  2^a-by{a-\-b) 

I4X'  —  'JX  2X  — I 

I2:v3  +  24:x;^  '  x'  +  2X  ' 


Summary 

Multiplying  both  the  numerator  and  the  denominator  of  a 
fraction  by  the  same  number  does  not  alter  the  value  of  the 
fraction. 

Dividing  both  the  numerator  and  the  denominator  of  a 
fraction  by  the  same  number  does  not  alter  the  value  of  the 
fraction. 

Fractions  having  the  same  denominator  are  added  or  sub- 
tracted by  dividing  the  sum  or  the  difference  of  the  numerators 
by  the  common  denominator. 

Fractions  having  different  denominators  are  added  or  sub- 
tracted by  first  reducing  them  to  equivalent  fractions  having 
a  common  denominator  (usually  the  least  common  denomina- 
tor) and  then  adding  or  subtracting  the  resulting  fractions. 

Fractions  are  multiplied  by  multiplying  their  numerators 
for  the  numerator  of  the  product,  and  their  denominators  for 
the  denominator  of  the  product. 


274  First-Year  Mathematics 

Fractions  are  divided  by  multiplying  the  dividend  by  the 
inverted  divisor. 

Cancelling,  or  removing,  the  same  factor  from  both  the 
numerator  and  the  denominator  of  a  fraction  is  equivalent  to 
dividing  both  the  numerator  and  the  denominator  by  the  same 
number. 

The  numerator  and  the  denominator,  when  spoken  of 
together,  are  called  the  terms  of  the  fraction. 


CHAPTER  XIII 
FACTORING.     QUADRATICS.     RADICALS 

198.  The  factors  of  a  number  are  its  exact  divisors.  The 
process  of  factoring  is  therefore  inverse  to  the  process  of  multi- 
plication. In  this  chapter  we  consider  only  factors  which  are 
non-fractional  (integral)  and  which  do  not  involve  radical 
signs  (i.  e.,  which  are  rational  numbers).  We,  therefore,  admit 
only  those  factors  of  a  number  which  divide  it  exactly  with  a 
rational  integral  quotient. 

A  prime  number  is  a  number  which  has  no  factors  except 
itself  and  unity.     Give  examples. 

A  prime  factor  is  a  factor  that  is  a  prime  number. 

A  number  has  only  one  set  of  prime  factors. 

Monomial  Factors 

1.  a{b-{-c)=ab+ac,  therefore  the  factors  of  ab-{-ac  are  a 
and  b+c.  The  monomial  a  appears  as  a  common  factor  of  the 
terms  of  ab-k-ac. 

2.  x^y{k-\-l-\-m)=kx^y-\-lx^y-\-mx^y,  therefore  the  factors 
of  kx^y-\-lx^y-\-mx^y  are  x^y  and  k+l+m.  The  monomial 
x^y  appears  as  a  common  factor  of  the  terms  of  the  given 
expression  kx^y-\-lx^y-\-mx^y. 

3.  Multiply  ;^x^y^  and  {2xy — ^x^  —  ^y^) .  What  are  the 
factors  of  6x^y'*—gx^y^  —  i^x'y^?  How  does  the  monomial 
^x'y3  appear  in  the  expression  Gx^y^  —  gx^y^  —  i$x^y^  ? 

4.  Factor  2,a^b  —  i2ab^. 

Since  306  is  a  factor  of  each  term,  it  is  a  factor  of  the  whole  expres- 
sion (each  term  in  succession).  We  obtain  the  quotient  a  — 46.  Then  306 
and  a— 46  are  the  factors  of  ^a^b~i2ab^,  or  ^a'b—  I2ah^  =  ;^ab(a  —  4b). 

Test  by  multiplication:  a— 4b 

3fl^ 
^a'b—i2ab' 

27s 


276  First- Year  Mathematics 

199.  An  expression  is  eqiud  to  the  product  of  all  of  its  prime 
factors  for  all  values  of  its  letters.  It  will  be  recalled  that  such 
equality  is  called  identity.    (See  p.  213.) 

By  this  principle,  we  can  test,  or  verify  the  correctness  of 
the  factors.     If  the  factors  of  T,a^b  —  i2ab'  are  ^ab  and  a  — 4b, 

then 

^a'b  —  1 2ab'  =  T,ab{a — 46) , 

for  all  values  of  a  and  b.     The  two  expressions  are  therefore 
equal  for  the  values,  a  =  5  and  6  =  1. 
Test  by  substitution: 
For  a  =  5  and  6  =  1; 

3a*6-i2a6^=3X5'Xi  — i2X5Xi'  =  i5 
and  3a6(a-46)  =3X5X1  (5-4)  =  15- 
Both  ^a'b  —  i2ab'  and  306 (a— 4^)  being  15  for  a  =  5  and  6  =  1, 
the  expression  is  correctly  factored. 

Verify  by  substituting  other  values  for  a  and  b. 

Exercise  LVII 

200.  Factor  the  following  expressions  and  test  results  both 
by  multiplication  and  by  substitution,  doing  as  many  as  you 
can  mentally: 

2.  am'-\-bm' 

3.  abc—abd 

4.  3njc4-6nj 

5.  2cdl+4cdm 

6.  ax'y-\-bx'y 

7.  cx'+dx3-\-fx^ 

8.  $xyz  +  i2X'yz-\-gxy'z' 

9.  $a^b  +  2^a'*c  —  ioa5d 

10.  i2a'b'r-{-iSa3b'^s  +  6a'^bH 

11.  Sx3y^-\-4x''y3 

12.  ^x'y'  —  2xy—^xy3 


Factoring,  Quadratics,  Radicals  277 

13.  isa^x  —  ioa^y  +  ^a^z 

14.  32a3&3— <jft6 

15.  6x3y'-\-T^x^yi+;^xy* 

16.  Sx'y'  +  i6xyz  —  24X'y'z' 

17.  2a»6«+a363-f-aM 

18.  Sa3b'c^-4a^b3c3-\-a^b'c3 

19.  i5:x;*  +  2a!cj4-5>'^ 

20.  i6a*6'+48a^6  — i6a6+8a 

21.  2I»*5  +  35»5^+56»5 

22.  3gMS2;7 -1-26^^1;'°  — 9im"z;9 

23.  4^f-g'h3-6ofig3h  +  S4fgh' 

24.  6ax^>' — 8a:»;3  _  ^ay^  —  4axy 

25.  77W*«4_22W3wS-|-2-^^S^3— ^^^6^-> 

26.  5£;"+x2'' 

27.  iC*+:»<'"'"* 

28.  a^h^y^-a^-'hy 

29.  isas^ft^y  +  asa+'^fisy 

30.  x^'^y^"' —  T)X3"'y'^*^-^2X^"'y* 

31.  /^''+6/fc  +  '^-)fe6+^/a+^ 

32.  6w2''  +  3^»3'-+5^  — g/w^''+^W.''+S^. 


Exercise  LVIII 
201.  Reduce  to  lowest  terms: 

36  +  3c  %x3y^  +  4ai;»  j3 

5^  +  5<^  i4:x;'>'z  +  7:x;j"z 

ab+ac  ^a^b  —  ioab' 

o. 


&:x;+c:x:  "   i$abc  +  2oa'b^ 

6x+6y  2km'— 6bm' 

1 2a  +  12b  ;^kn3  —  gbn3 

^x'  —  ioxy  ^ad'  —  i$bd'  —  iocd' 


4ax—Say  '  6axy'  —  iSbxy'  —  i2cxy' 


278  First- Year  Mathematics 

Perform  the  operations  indicated  in  the  following  problems: 

SX'        2X--4  ^^       5     I      7 


5ic  — 10       6x'*  jf— 3     3^—9 

ab+ac    ab—ac  i       .       2 

13-  rr7T7  + 


bd—cd    bd+cd  ax+ay    bx+by    cx+cy 

T,m  —  ^n      gm—gn  2  3        5C+io— 2 

5w+5n  *  low  +  iow        '  ^x—Sy    x—2y       4^— 8v 


Exercise  LIX 

202.  Solve  the  following  equations: 

1.  Solve  nix+nx=am-\-anior  x. 

Factoring,  x(m  +  n)=a{m  +  n). 
Thena;  =  a.     Why?     Check. 

2.  ax+bx=ac-{-bc.     Solve  for  jc 

3.  ^a^y—2,b^y  =  \oa'^m—6b^m.     Solve  for  )> 

4.  kv-\-lv  —  T,mv=^kr^s-^^ir^s  —  iz,mr^s.     Solve  for  w 

5.  ak+bm—am-\-bk.     Solve  for  ^ 

6 .  abx—abd=cx—cd.     Solve  for  rv 

7.  mt—6tw^  =  ^ms  —  T)Osw^.     Solve  for  w 

8.  ax-\-bx=cx-\-d.     Solve  for  :x; 

o.  a-\-b= — I — .     Solve  for  y 

y    y 

,  ,  abn'^  .  abm^       ^  ,      ,      , 

10.  m'-\-—r-=n'-\ — 7—  .     Solve  for  k 

a 

11.  u-\-v  —  -.     Solve  for  ic. 

X 

203.  Solve  the  following  problems: 

1.  In  5  quarts  of  a  certain  solution  there  is  m  times  as 
much  water  as  chemicals.  How  much  water  and  how  much 
chemicals  in  the  solution  ? 

2.  In  an  alloy  of  two  metals,  the  ratio  of  the  weight  of 
one  metal  to  the  weight  of  the  other  is  T.k.     There  are  3  lb. 


I 


Factoring,  Quadratics,  Radicals  279 

more  of  the  first  metal  than  of  the  second  in  the  alloy.     Find 
the  weight  of  each  in  the  alloy. 

3.  Two  numbers  have  the  ratio  a:h,  and  their  sum  is  c. 
Find  the  numbers. 

4.  Two  numbers  have  the  ratio  m:n,  and  their  difference 
is  equal  to  the  sum  of  u  and  v.     Find  the  numbers. 

5.  The  second  of  three  numbers  is  r  times  the  first,  and  the 
third  is  5  times  the  second.  Find  the  three  numbers,  if  their 
sum  is  equal  to  the  sum  of  the  numbers  a,  ar,  and  ars. 

204.  Polynomials  whose  terms  may  be  grouped  to  show 
a  common  binomial  factor. 

1.  Multiply  0+6  by  w+w. 

a+h 
m+n 
am-\-bm+an-\-bn. 
In  this  multiplication  a-{-b  is  first  multiplied  by  m,  then  by  n 
and  the  products  are  added.     Therefore, 

(a-\-b)  (w+«)  =  m(a+6)+»(a+&)  =am+bm-]-an+bn. 

2.  Factor  3a  +  3&  + 5<J  +  5&. 

3a+3b  +  5a^5b=3(a-\-b)-{-5(a  +  b)  =  ia+b){3  +  5). 
This  method  of  factoring  is  called  grouping. 

3.  FsLCtoT  ac  +  bc+ad+bd. 

ac+bc-{-ad  +  bd=c{a-\-b)-\-d(a  +  b)  =  {a+b){c+d). 

Test  by  multiplication  : 

c-\-d 

a+b 

ac-\-ad-{-bc+bd 
Test  by  substitution: 

Let  0  =  1;  6  =  2;  c  =  3;  d=4. 
Then  ac -\-bc-\-ad+bd  =  i- 3-^ 2 -3  +  1' 4  +  2- /^  = 
3+6+4+8  =  21 

and,  (c+J)(a  +  fe)  =  (3  +  4)(i+2)=7X3  =  2i. 
Therefore,  ac+bc+ad  +  bd={c+ d) (a + 6) . 


28o  First-Year  Mathematics 

4.  Factor  I4JC3  — 6:x;*  — 2i:x;+9. 
Test  by  substitution  and  by  multiplication. 

Exercise  LX 

205.  Resolve  into  factors  the   following  expressions  and 
test  results,  doing  as  many  as  you  can  mentally: 

1.  ax-{-bx+am+bm  15.  ^ac+;iax  —  $c  —  ^x 

2.  ar+br-\-as-\-bs  16.  9  — 15^  +  27/-^— 45^3 

3.  ad+bd+at+bt  17.  8gh  +  i2ah  +  iobg-\-i$ab 

4.  ^a+^b+ay+by  18.  i^z—6  —  2ozw+Siv 

5.  ak—bk+al—bl  19.  2m'  +  7,km  —  i4mn  —  2ikn 

6.  ax^— 6jc^+ay3— ftj3  20.  3ax4-3aft  +  2X^+26x+6+x 

7.  a&c4-a6x+»c+»jc  21.  ^^+4X—4X^z—4z 

8.  a^yfe+a^/+6^^+6^/  22.  i+r— r^'jcy— r^^v); 

9.  ^au  —  $av+mu  —  mv  23.  x*— jc^+i— a; 

10.  m^a  +  nia'-\-in^a'  +  m^a^  24.  (a  +  m)(c+»)— 2«(a+w) 

11.  a^—ad+ab—bd  25.  (5c;+>')(a+6)— (:x;+>')(6+c) 

12.  a;'^+5x4+:x;3+5:x;  26.  w(:x;+)')^  +  («;+3') 

13.  6x^—gx  —  ioxy  +  i^y  27.  a-^+'"+a'"6''+a-^6'"+&'""''^ 

14.  2w?3  +  w^+6m  +  3  28.  2:)£;''+'— 3:i;>^  — 2:x;'';y+3y+' 

29.  3w"+2— 5w"n3  4-3m^n^— 5»^+3 

Exercise  LXI 

206.  Reduce  to  lowest  terms: 
ax+bx+am+bm  ax'—bx'-\-ay^—by' 


I. 


ar+br+as+bs  ^'  mx^  -\-my^  -\-nx^  +ny^ 

T,u—yo-\-au—av  x'*  — 2X3  +  7:!C  — 14 

4- 


Sbu  —  ^bv  +  2ku  —  2kv  '   2x3— 4:)£;^  +  6:x;  — 12 

2iw—3a^  — 106^  +  156/ 
loaw  —  1 5a/ —  26m  +  36/ 


Factoring,  Quadratics,  Radicals  281 

Perform  the  indicated  operations  in  the  following: 
30—36      cx—dx-\-cy—dy 


6. 


5c  — 5(i     am~bm  +  an  —  bn 
x^  —xy—6x  +  6y     ^ac  —  ^bc 
ax  —  bx—ay-\-by      xy—dy 

+ 


ab+ac—bk  —  ck  bm+cm+bk+ck 
2am  —  2bm—^an-\-^bn  _  ^ax—^bx 
2ctn  —  ^cn  +  2dm  —  ^dn  '  6cy  +  6dy  ' 

Exercise  LXII 
207.  Solve  the  following  equations: 

1.  ax+bx=ad+bd+ac+bc.     Solve  for  a; 

2.  ^a'bx-\-i2bcd=4ab^x+gacd.     Solve  for  ^ 

3.  ax—bc—ac=ab-\-bx.     Solve  for  .r 

4.  am+bm=;^am  +  ^bm+ak+bk—an—bn.     Solve  for  w 

5.  av+bv—;icv  =  ^a+^b  —  i^c-\-2aw-{-2bw—6cw.        Solve 
for  V.     Solve  for  w 

6.  a—b= — I ■.     Solve  for  7 

y    y    y 

b     d     bd     c     be       r,  ,      , 

7.  i-\ = 1 — I .     Solve  for  ic 

a    x    ax    X    ax 

bm      2     am       2 

6.  -r-A — 7= — ,H .     Solve  for  m 

ocd     50     2ca     15a 

Solve  for  u 


2U-\-l       4U+2 


■o.      Solve  for  x 


x^+x'+x  +  i      X^—^X^+X—T, 

X  2  x 

II. 1 1 — - — =4.      Solve  for  V. 

2J-2     3>'-3     Ay- A 

208.  Solve  the  following  problems: 

I.  In  one  pile  of  brick  there  are  r  times  as  many  bricks 
as  in  a  second  pile.     If  placed  in  three  piles  of  a,  r,  and  ar 


283 


First-  Year  Mathematics 


bricks  respectively,  there  is  one  brick  left  over.     How  many- 
bricks  in  each  of  the  two  original  piles  ? 

2.  In  an  alloy  of  two  metals,  the  ratio  of  the  weights  of 
the  metals  is  a:b.  A  certain  mass  of  the  alloy  is  a+b  pounds 
more  than  another  mass,  and  the  weight  of  the  two  masses 
together  is  3(1+36.  How  many  pounds  of  each  metal  are 
there  in  each  of  the  masses  of  alloy  ? 


The  Trinomial  Square 

Exercise  LXIH 

209.  Multiply: 

I.  (a+ty 

8.  {m-ny 

IS- 

(3w3-5n)» 

2.  {r+sy 

9.  {x  +  sY 

16. 

{ab  +  T^x'^y 

3.  (c+dy 

10.  {y-^y 

17- 

(a'm  —  $x3y 

4.  {a-by 

II.  {T-ay 

18. 

{abc3-\-2X''y) 

5-  ig-hy 

12.  {2u—T,vy 

19. 

{x'y  —  'jxy^y 

6.  {r-sy 

13.  {Ab^-iyy 

20. 

{y^si-4rs^) 

7.  (c-dy 

14.  {zx^  +  2yy 

21. 

{x-\-i/2y. 

22.  From  these  exercises  make  a  rule  for  squaring  a  binomial. 

When  a  binomial  is  multiplied  by  itself  the  result  is  a  tri- 
nomial which  consists  of  the  square  of  the  first  term  of  the 
binomial,  plus  or  minus  (according  as  the  binomial  is  a  sum 
or  a  diflference)  twice  the  product  of  the  first  and  second  terms 
of  the  binomial,  plus  the  square  of  the  second  term  of  the 
binominal. 

Then  any  trinomial,  in  which  two  terms  are  squares  (and 
positive)  and  the  other  term  is  plus  or  minus  twice  the  product 
of  the  square  roots  of  those  two  terms,  is  the  square  of  the 
sum  or  difference  of  those  two  square  roots  according  as  the 
third  term  is  plus  or  minus. 

23.  FsictoT  k^'+ekl+gl'. 

k'  and  gl'  are  squares,  and  6kl  is  twice  the  product  of  their 
square  roots,  therefore,  k'-ir6kl+ gl'  =  {k-\-^iy. 


Factoring,  Quadratics,  Radicals 


283 


Exercise  LXIV 

210.  Factor  the  following  expressions,  and  test  each  re- 
sult, doing  as  many  as  you  can  mentally : 


1.  x'-\-2xy-\-y^ 

2.  a^  — 2a6  +  6* 

3.  m'+^mn+^n' 

4.  4a^— 4a-|-i 

5.  b^-6b+g 

6.  ;fe^-|-8)fe4-i6 

7.  a^-hioab-\-2sb^ 

8.  4m^  —  i2am  +  ga^ 

9.  25-t-8or  +  64r^ 


13.  49  — i4ow*  +  ioo»'' 

14.  a'b'c^+Sabc  + 16 

15.  49X*^  — I54:x;3^  +  i2i;y2 


16.  625^4 -|-5ow^z;*-|-'y4 

17.  i6yfe^-l-72yfe'yw+8i)'^w' 

18.  169^2-78^/^+9/4 

19.  2255"— 4205*^C5  + 196c' ° 

20.  (w  +  »)2  +  2(w  +  »)+I 

21.  (m+i;)2+4/(«+2;)+4/2 

10.  i2ia^  +  igSay-\-8iy'    22.  9)fe2+6^(r+5)  +  (r+5)* 

11.  c'-i6c+64  23.  (a+6)2  +  2(a+&)(c+<i)  +  (c+J)=' 

12.  5c;4+3ox*  +  225  24.  m^  +  2mn -\-n'  + 2m -\- 271  +  1 

25.  w2  +  2mw+«^+6aw  +  6aw  +  9a* 

26.  a^+b^-\-c'  +  2ab  +  2ac-\-2bc. 

27.  Square  53. 

53' =  (50  +  3)' =  50' +  2- 50- 3 +3' =  2,500+300  +  9  =  2,809. 

28.  Square  68. 

682  =  (7o  — 2)2=4,900  — 280+4  =  4,624. 

29.  Square  the  following  numbers  mentally: 

(i)  13  (5)  22  (9)  43  (13)  91 

(2)  14  (6)  -31  (10)  38  (14)  89 

(3)  15  (7)  19  (11)  72  (15)  67 

(4)  21  (8)  18  (12)  81  (16)  103. 

Exercise  LXV 

211.  Factor  the  following  by  any  method,  and  test: 

1.  a'ki-2abk^-{-¥k^ 

2.  X'*y+x'y3—x3y^—xy* 

3.  ax^+ax'+ax+a 

4.  76:v*+426:!C)'  +  636)'2 


284 


First-  Year  Mathematics 


5.  i2a''xy-{-i2axy^--^xy 

6.  5Ji(;s  +  S:»*  +  4^^+4^^+3:x;+3 

7.  {c^-d){c' -\-d')  +  2c^d-\-2cd' 

8.  x'—6xy-\-gy^  —  2Xz-\-6yz-^z^ 

9.  ga'm  +  ga^n-^i2am  +  i2an  +  4m-\-4n 
10.  (^4->')='(3i;-3')-(3(;-3')='(3f+>'). 

Quadratic  Expressions  and  Qaadratic  Equations 

212.  An  expression  in  which  the  highest  power  of  the 
unknown  is  the  second  power  is  called  a  second-degree  ex- 
pression, or  a  quadratic  expression. 

213.  An  equation  in  which  the  highest  power  of  the  un- 
known is  the  second  power,  is  called  a  second-degree  equation 
or  a  quadratic  equation. 

For  example,  :x:^  — ioai;-f-2i  and  )'^  — 25  are  second  degree, 
or  quadratic,  expressions;  while  x'  —  iox-{-2i=o,  and  ;y^  — 25 
=0  are  second  degree,  or  quadratic,  equations. 

214.  Graph,  on  cross-Uned  paper,  the  quadratic  expres- 
sion, ic^  — iax;-|-2i. 


--77 .    

t                    1 

3                        I 

__~_l E_    _    - 

-~t                  2j 

-X                    t 

A                  S 

zz ii^~ 

A                t  ' 

tff     <i' 

X,--j^,££lI^_^ 

Z  ■>L3-iiTiSW'~T^T*io7 

-  -    ^i 

X 

.r'  — IO.V+2I 

0 

+  21 

+  I 

+  12 

+    2 

+  5 

+  3 

0 

+  4 

-  3 

+  5 

-  4 

+  6 

-  3 

+  7 

0 

+  8 

+  5 

+  9 

+  12 

+  10 

+  21 

Fig.  238 

Plot  the  points  of  the  table  and  draw  through  them  a  smooth 
curve.    The  scale  of  Fig.  238  is  horizontally,  i  space  =  1,  and 


Factoring,  Quadratics,  Radicals  285 

vertically,  i  space  =  2.  What  is  the  value  of  rv^  — io:x;  +  2i  at 
the  point  A  ?    At  B  ?    At  C  ?    At  G  ?    At  D  ?    At  H  ? 

What  is  the  value  of  x  at  the  point  A  ?  At  B  ?  At  C  ? 
At  G  ?    At  D  ?    At  H  ?    At  I  ? 

The  vertical  distance  of  any  point  from  the  horizontal 
axis  represents  the  value  of  the  expression,  x^  —  iox-\-2i,  at 
that  point. 

The  horizontal  distance  from  o  to  the  foot  of  a  perpendicu- 
lar to  the  horizontal  through  any  point  of  the  curve  represents 
the  value  of  x  for  that  point. 

The  lines  O  X  and  O  Y  of  Fig.  238  are  called  axes. 

215.  Graph  the  following  quadratic  expressions  between 
the  indicated  limits  for  x  and  keep  the  graphs  for  reference : 

1.  x^—^x  +  i2,  from  o  to  +8 

2.  ^x;^— 6:x;  +  5,  from  — i  to  +7 

3.  x^  —  ^x  —  d,  from  —2  to  -f; 

4.  x^—2,x  —  io,  from  —3  to  +6 

5.  51;^  — io:x;-|-24,  from  +2  to  -|-8 

6.  :x;^  — iox-f25,  from  +2  to  +8 

7.  4:x;^  — i2:x;  +  5,  from  —2  to  +5 

8.  4X*-l-8x  — 5,  from  —4  to  +2. 

9.  Show  again  on  each  of  the  graphs  the  points  of  the 
curve  where  the  expression  is  o.  Give  the  values  of  x  for 
these  points. 

216.  The  equation,  :x;^  — io:x;  +  2i  =0,  may  be  regarded  as 
made  by  placing  the  expression,  x^  — io:x;-f  21,  equal  to  zero. 
To  solve  the  equation,  :x;^  — io:)C  +  2i  =0,  means  to  find  the 
value  or  the  values  of  x  for  which  the  expression  is  o.  On 
the  graph  this  means  to  find  how  far  it  is  from  the  o-point, 
called  the  origin,  to  where  the  curve  intersects  the  horizontal. 

The  vertical  drawn  upward  through  the  lowest  point,  F, 
of  the  curve  divides  the  curve  into  two  symmetrical  parts.     If 


a86  First-Year  Matliemalics 

the  paper  were  folded  over  F  N  as  a  hinge-line  the  parts  of 
the  curve  would  coincide. 

This  line,  F  N,  is  called  the  axis  of  the  curve. 

The  axis  crosses  the  horizontal  at  the  point  +5,  which  is 
midway  between  the  points  +3  and  +7  that  represent  the 
two  values  of  x  in  the  equation,  a;'  — icx)c-f2i=o. 

217.  From  the  values  of  x  given  in  answering  problems  i 
to  8  (§215),  give  exact,  or  approximate,  values  of  x  that  will 
satisfy  the  following  equations: 


I. 

x''—%x->ri2=o 

5- 

:»*  — io:)!;  +  24=o 

2. 

x'—6x-\-  5=0 

6. 

:»*  — io:>;-|-25=o 

3- 

x^  —  <^x—  6=0 

7- 

^'  —  i2X-\-  5=0 

4- 

x'—2^x  —  io=o 

8. 

4JC^-}-  %x—  5=0. 

9.  Graph  the  expression,  x^—^,  and  from  the  graph  give 
the  values  of  x  that  will  satisfy  the  equation,  :x:^  —  4=0.  Check 
by  substituting  in  the  equation. 

10.  Graph  the  expression,  iojc;^  +  2ix  — 10,  for  values  of  x 
between  —4  and  +2  and  find  from  the  graph  the  approximate 
values  of  x  that  satisfy  the  equation,  ioj(;^  +  2i:x;  — 10=0. 

Quadratic  Equations  Solved  Algebraically 

218.  The  equations  that  have  been  solved  graphically, 
sometimes  exactly  and  sometimes  only  approximately  in  the 
foregoing  article,  may  be  solved  exactly  and  more  easily  by 
algebraic  methods.  Furthermore,  algebraic  methods  furnish 
exact  solutions  for  all  kinds  of  quadratic  equations. 

Exercise  LXVI 

219.  Add  a  third  term  to  make  squares  of  the  following  and 
give  the  square  roots  of  the  resulting  trinomials: 

1.  tn^+n'  5.  m^+6mn 

2.  m'  +  2mn  6.  n'+Smn 

3.  2mn+n'  7.  x^+4X 

4.  m^-\-^n'  8.  y'+6y 


Factoring,  Quadratics,  Radicals  287 

9.  m^  +  iom  15.  ga'b'+4c' 

10.  k^+^k  16.  gx'  +  ^ok^x 

11.  X'+^X  17.    4X^  +  12X 

12.  r*  +  7r  18.  4^^  +  ia» 

13.  x'  +  bx  19.  16^* +  56^^ 

14.  r'  +  ys  20.  4x^  +  ii:x;. 

21.  Make  a  rule  for  adding  a  third  term  to  complete  the 
square,  (i)  when  the  two  square  terms  are  given,  (2)  when 
one  square  term  and  the  cross-product  term  are  given. 

220.  Solve  algebraically  for  :x;  the  equation,  :»*  — io:x;  +  2i  =0. 

Write  the  equation  thus: 

at:'— ioA!r=  —  21. 
Add  25  to  both  sides  to  make  the  iirst  side  a  trinomial  square: 
x'  —  iox  +  2^=4,    or 
(x-5)»=4. 
Take  the  square  root  of  both  sides,  remembering  that  4  has  two 
square  roots,  +2  and  —2;   thus: 

x—';  =  ±2. 

Square-root  Axiom. — Any  number  has  two  square  roots  of  the  same 
absolute  value,  but  of  contrary  sign. 
Using  the  first  sign, 

jc— 5  =  2,  whence  ic=  +7. 
Using  the  second  sign, 

a:— 5  =  —  2,  whence  x=  +3. 
Check  both  values  of  x  by  substituting  in  the  equation,  »» -■  xox  + 


I.  Show  that  ic^  — io^-|-2i  =0  is  the  algebraic  statement  of 
the  verbal  problem : 

The  area  of  a  rectangle  10  units  long  is  21  square  units 
greater  than  the  area  of  a  square  whose  side  equals  the  un- 
known dimension  of  the  rectangle. 

The  algebraic  method  just  given  is  called  the  method  of 
solving  the  equation  by  completing  the  square. 


288  First-Year  Matliematics 

2.  Solve  the  first  six  exercises  of  §217  by  the  method  of 
completing  the  square. 

3.  Solve  the  equation,  v^+S.v  — 36,  by  the  method  of  com- 
pleting the  sqnare. 

y+l  =  ±^ 

y=     \      -4,    or  -9. 

4.  Show  that  >'*  +  5>'— 36=0  is  the  algebraic  statement  of 
the  verbal  problem : 

One  dimension  of  a  rectangle  is  5  units  and  the  other  is 
equal  to  the  side  of  a  square.  The  sum  of  the  areas  of  the 
rectangle  of  the  square  is  36  square  units.  Find  the  unknown 
dimension  of  the  rectangle. 

221.  Though  a  quadratic  equation  generally  has  two  solu- 
tions, this  does  not  mean  that  every  problem  that  leads  to  a 
quadratic  equation  has  two  solutions.  The  nature  of  the 
conditions  of  the  problem  may  be  such  as  to  make  one,  or 
even  both  of  the  solutions  of  the  quadratic  impossible,  or 
inappropriate  or  meaningless.  When  neither  of  the  two  solu- 
tions of  the  quadratic  is  a  solution  of  the  problem  it  usually 
means  that  the  conditions  of  the  problem  are  impossible,  or 
are  contradictory,  or  that  the  problem  is  erroneously  stated. 
To  decide  which  solution,  if  either,  meets  the  conditions  stated 
in  a  problem,  it  is  necessary  to  substitute  the  solutions  in  the 
conditions  of  the  problem,  and  to  reject  solutions  of  the  equa- 
tion which  do  not  meet  the  conditions. 

The  graphical  method  used  with  the  quadratic  expressions 
of  problems  7  and  8,  §215,  and  of  problem  10,  §217,  p.  286, 
furnished  only  approximate  values  of  x  for  the  corresponding 
quadratic  equations.  The  algebraic  method  furnishes  exact 
solutions  for  quadratic  equations. 

I.  Solve  the  quadratic  equation,  4JC*  — 12^-1-5=0. 

By  the  Div.  Ax.  the  given  equation  may  be  written 
x'  —  ;ix  +  ^=  o. 


Factoring,  Quadratics,  Radicals 


289 


By  the  Add.  Ax.  this  equation  gives 

or,         (3C- §)»  =  !. 
By  the  Sq.-rt.  Ax. 

whence,     x  =  %±,i 
and,  »  =  I,  or  i. 

Check  by  substituting  in  the  given  equation. 

2.  Solve  the  equation,  4x^-\-%x  —  ^='0,  and  check. 

3.  Solve  the  equation,  iox*  +  2ix— 10=0,  and  check. 

4.  Solve  and  check  the  following  quadratics: 

(i)  6:x;^  — lyrx;  — 14=0        (4)     45^+455  —  36=0 

(2)  6:x;^  +  7:!C  — 20=0  (5)  105^  —  215  +  10=0 

(3)  9:x;^ +  30:^-24=0         (6)  125^—715+42=0. 

5.  Solve  the  following  quadratic  equations  by  the  method 


of  completing  the  square,  and  check: 


(l) 

x*  +  25£;— 3=0 

(2) 

3i;^+4:x:  +  3=o 

(3) 

a;»+4:c— 5=0 

(4) 

a^+8a  — 20=0 

(5) 

>'*  +  i4>'+45=o 

(6) 

/'  +  i4/  +  5i=o 

(7: 

ife«-8s  =  i2yfe 

(8) 

z' =  102  +  24 

(9) 

W^+QI  =6w 

(10] 

f»+3r+2=o' 

.    (11] 

w*  +  5w+6=o 

(12)  h'-\-j\o  =  i^h 

(13)  X'+X=42 

(14)  x'-\-6mx-\-5m'=o 
x""     X 

(.5)  -~-=-i 

Z  Z  Z  — I 


(18) 


/+8  3^+4 
^+1  ^+3^ 
k-\-2     k  +  4 


222.  Solve  the  following  problems: 

I.  The  length  of  a  rectangular  field  is  4  yards  more  than 
the  width  and  the  area  is  60  square  yards.  Find  the  dimen- 
sions. 


290 


First-Year  Mathematics 


2.  The  h)rpotenuse  of  a  right  triangle  is  10  ft.  and  one  of 
the  sides  is  2  ft.  longer  than  the  other.  Find  the  length  of 
the  sides. 

3.  The  sum  of  the  areas  of  two  square  fields  is  61  square 
rods,  and  a  side  of  one  is  i  rod  longer  than  a  side  of  the  other. 
Find  the  sides  of  both  squares. 

4.  What  must  be  the  dimensions  of  a  coal-bin  to  hold 
6  tons  of  coal,  if  the  depth  is  6  ft.  and  the  length  is  equal  to 
the  sum  of  the  width  and  depth,  allowing  40  cu.  ft.  of  space 
per  ton  of  coal  ? 

5.  Telegraph  poles  are  placed  at  equal  distances  along  a 
railway.  In  order  that  there  be  two  less  per  mile  it  would  be 
necessary  to  increase  by  24  ft.  the  distance  between  every  two 
consecutive  poles.     Find  the  number  of  poles  to  the  mile. 


Quadratics  Leading  to  Irrational  Numbers 

223.  To  solve  quadratics  of  the  kind  to  be  considered 
below  a  new  difficulty  is  met. 

I.  Graph  the  expression,  w^+6w  +  2,  between  values  of  m 
from  —8  to  +2. 


m 

m'  +  6m  +  2 

-8 

+  18 

-7 

+  9 

-6 

+  2 

-5 

-  3 

-4 

-  6 

-3 

-  7 

—2 

-  6 

—  I 

-  3 

0 

+  2 

+  1 

+  9 

4-2 

+  18 

Factoring,  Quadratics,  Radicals 


291 


:i^^;s:iii::zi::ii:i:i:ii::iii 

Illlllllllll^v^Iallllllllllllll 
Z  ———1Z~~—~ 5^  — —  —  —  —  —  — —  — 

:iii::iiiiiiii:iiiii:iii'5§iiii 

iiilliil 

^<::-i III_I 


Scale. — The  unit  for  w  is  i  large  square,  and  for  m'  +  6m  +  2, 
I  small  square. 

Plot  the  points  of  the  table  to  the  indicated  scale,  and  draw 
through  them  a  smooth  curve. 

What  is  the  value  of  w^+6m  +  2  at  the  point  A?  At  B  ? 
At  C  ?    At  D  ?    At  H  ?    At  E  ?    At  F  ?    At  G  ? 


292  First-Year  MaUiematics 

What  is  the  value  of  m  at  the  point  A  ?  At  B  ?  At  C  ? 
At  D  ?    At  H  ?    At  E  ?    At  F  ?    At  G  ? 

The  vertical  distance  of  any  point  of  the  curve  from  the 
the  horizontal  axis  represents  the  value  of  the  expression, 
m'-\-6m-\-2,  at  that  point. 

What  distances  on  the  drawing  represent  the  values  of  m 
for  any  point  of  the  curve  ? 

2.  Graph  the  following  quadratic  expressions  between 
values  of  m  from  —8  to  +2. 

(i)  m^-\-6m+4  (3)  m'-\-6m-\-g 

(2)  m^  +  6m-\-6  (4)  m^+6m  +  io. 

The  equation,  w^+6w  +  2=o,  is  made  by  placing  the  ex- 
pression, m^+6m-\-2,  equal  to  zero.  Show  on  the  curve  of 
Fig.  239  the  values  of  m  for  the  points  where  m^  +6m-\-2  equals 
zero. 

H  L  is  the  axis  of  the  curve.  It  crosses  the  horizontal 
at  K.     Read  from  the  drawing  the  lengths  of  K  C  and  K  F. 

Show  that  one  value  of  m  which  gives  m^-\-6tn  +  2  equal  to 
zero,  is 

OC  =  OK-KC, 

and  that  the  other  value  of  m  is 

OF  =  OK-fKF. 

Show  from  the  graph  of  Fig.  239  that  if  the  two  values  of 
m  in  m'-\-6m-\-2=o  be  denoted  by  w'  and  m'\  they  may  be 
represented  thus, 

and,  w''=— 3— w 

where  n  denotes  the  same  number  in  both  values,  and  (2)  that 
n  is  some  number  between  2 . 5  and  3  units. 

224.  The  algebraic  solution  of  the  equation,  m^-\-6m-\-2=o, 
will  show  exactly  what  this  number,  n,  of  §223,  is,  thereby 
furnishing  both  values  of  m  exactlv. 


Factoring,  Quadratics,  Radicals  293 

1.  Solve  algebraically  for  m  the  equation,  m^-\-6m-\-9.=o. 
Write  m'+6m=—2 

Add  9,  w^+6w+9  =  7 

or,  (w+3)^=7. 

Taking  square  roots,  w  +  3  =  ±]/7. 

Whence  w  =  —  3  ±  1/7. 

The  two  values  of  m  are  then 

w'= -3  +  1/7 
and  w"  =  —  3  — 1/7 . 

The  number,  n,  used  above  to  denote  the  distances  in 
Fig.  239  from  K  to  C  and  from  K  to  F,  is  1/7. 

2.  Solve  the  following  equations  algebraically,  and  point 
out  on  the  graphs  of  problem  2  (§223)  the  lines  that  represent 
the  numbers  used  in  expressing  the  results: 

(i)  m^-\-6m+4=o  (3)  m'+6m+g=o 

(2)  w^+6w+6=o  (4)  m^+6m  +  io=o. 

3.  Solve  the  following  quadratic  equations  algebraically: 

(i)  ;x;^  +  iox  +  20=o  (6)  :x;^ +  14:^+42=0 

(2)  ;x;^  +  io:x;4-22=o  (7)  :x;^  +  i 4:^+44=0 

(3)  ^^  +  i2:x;+33=o  (8)  x'  +  i6x+62=o 

(4)  a;^  +  i2:x;  +  29=o  (9)  :x;^— 6^+4=0 

(5)  ^^  —  14^+41=0  (10)  x'—Sx  +  $=o. 

Radicals 

225.  Numbers  like  1/7,  1/5,  and  1/3  can  be  expressed 
only  approximately  without  using  the  radical  sign.  Such 
numbers  are  called  irrational  numbers,  or  radicals. 

In  the  graphical  solution  of  problem  i,  §223,  one  point  at 
which  the  curve  cuts  the  horizontal  is  three  units  to  the  left 
of  the  origin  (—3),  and  back  to  the  right  approximately  2^ 
units,  or  exactly  1/7  units.  That  is,  one  point  of  intersection 
is  expressed  exactly  by  —3  +  1/7. 


294 


First-Year  Matliemalics 


The  second  point  is  three  units  to  the  left  of  the  origin 
(—3),  and  2 J  additional  units  approximately  (]/'7  units  ex- 
actly) in  the  same  direction  (—2 J,  or  —1/7).  The  second 
point  of  intersection  is  expressed  exactly  by  —3  —  1/7. 
Point  out  in  Fig.  239  the  line  that  represents  -\/^. 
Point  out  on  the  graph  of  problem  3  (i)  (p.  293)  the  line 
that  represents  1/5. 

Point  out  the  line  that  represents  1/3  in  problem  2  (2) 
(P-  239)- 

226.  To  show  in  a  different  way  a  line  whose  exact  length 
can  be  expressed  only  by  a  radical,  draw  a  right  triangle 
having   a   base   3   inches   and    hypotenuse   4 
inches;   calculate  the  length  a  of  the  altitude. 
a^  =  4^-3^=7 
a  =1/7. 
Therefore  the  exact   length  of  the  line  a 
is  expressed  by  1/7,  a  number  that   cannot 
be  expressed  exactly  without  the  root  sign. 
Again,  if  a  triangle  be  constructed  with  a  base  of  6  inches, 
and  two  sides  of  8  inches  each,  we  can  draw  the 
altitude  and  can  calculate  its  length,  a,  thus, 
a^=8^-3^  =  55. 
Therefore,  a  =  1/55  inches.     The  number 
1/55   cannot,  without   the    root   sign,  |/,  be 
expressed  as  a  number  of  units,  either  integral 
or  fractional. 

If  another  triangle  be  constructed  on  this 
same  base  with  sides  of  8  inches  extending  in 
the  opposite  direction  from  the  first,  what  will 
be  its  altitude  ? 

What  will  be  the  sum  of  the  two  altitudes, 
or  the  diagonal  of  the  quadrilateral  formed  ? 

This   sum   is   written   most   briefly   thus,    2 1/55 . 


240 


Fig.  241 


i 


Factoring,  Quadratics,  Radicals  295 

Approximating  the  Value  of  a  Radical 

227.  By  the  process  of  extracting  the  square  root  of  an 

arithmetical  number,  the  square  root  of  7,  i.  e.,  7/7,  is  2.64. 

But  this  result  is  not  exact ;  because  the  square  of  2 .  64  is  only 

6 .  9696,  which  falls  short  of  7  by  a  Uttle 

7.0000  I  2.64      jjjQj.g  ^^^^     Q^      g^^  ^Yie  square  of  2.65 

is  7.0225,  a  number  larger  than   7   by 

£^  .02  +  .     So  1/7  is  between  2.64  and  2.65. 

If  the  root  be  found  to  one  more  deci- 


46 

524      2400 

2096  ^^^  place,  2 .645  is  obtained.    The  square 

^04  of  this  number  is  6.996025,  which  isless 

than   7   by  not   quite    .004.     While  the 

square  of  2.646  is  larger  than  7  by  more  than  .0003.     Then 

1/7  lies  between  2 .  645  and  2 .  646. 

Continuing  the  process  of  extracting  the  square  root  of  7, 

we  obtain  2.6457.     The  square  of  this  number  is  less  than  7 

by  only   .0003,  while  the  square  of  2.6458  is  larger  than  7. 

So  7/7  lies  between  2.6457  and  2.6458. 

Expressed  in  symbols: 

2.64     <  1/7  <  2.65 

2.645  <i/7< 2.646 

2.6457<  1/7  <  2.6458. 

Thus  by  continuing  the  process  of  extracting  the  square 
root  of  7  and  getting  additional  figures  we  may  approximate 
1/7  to  any  desired  degree  of  accuracy,  but  we  cannot  find  its 
exact  value.  1/7  expresses  the  exact  value  of  the  square  root 
of  7. 

In  similar  manner  show  that  "1/55  lies  between  7.4161 
and  7.4162. 

228.  Indicated  roots  which  can  be  found  only  approxi- 
mately are  classed  among  irrational  numbers.  When  written 
with  a  radical  sign,  they  are  called  radicals. 

Integers  and  fractions,   both  positive  and  negative,   are 


296  First-Year  Mathematics 

rational  numbers:    for  exampl  %   2,    —5,  I/4,  V\,  f ,  3.57; 
etc. 

229.  To  check  the  value  of  m  found  in  problem  i,  §224, 
substitute  (—3  +  1/7)  for  m  in  the  equation,  m"+6w  +  2=o. 

This  gives    (-3  +  l/7)*+6(-3  +  i/7)+2=o. 
Reducing     (_3  +  ,/7)^  =  (_3)»-t-2(-3)0/7)4-(i/7)'        (i) 
=  9-6^/7+7. 

{\^y  =  T,  because  the  square  root  of  a  number  when 
multiplied  by  itself  (or  squared)  produces  the  number.  Thus 
1/3X^3  =  3;  \/xX^/x=x. 

The  way  to  express  the  product  of  a  rational  number, 
as  6,  and  an  irrational  number,  as  \/'j,  is  to  write  the  rational 
and  irrational  numbers  side  by  side  thus:    61/7. 

The  product,  6y^7,  is  read:  "6  times  the  square  root  of  7." 

Equation  (i)  may  now  be  written: 

9—61/7  +  7  —  18+61/7+2=0 
9  +  7-i8  +  2+6v^7-6v^=o  (2) 

0=0. 

The  equation  then  is  satisfied  for  the  value  of  w=  —  3  +  1/7. 
Show  by  substituting  that  the  second  value  of  m,  —3  —  1/7, 
also  satisfies  the  equation,  w^+6m  +  2=o. 

Thus,  (-3-v/7)^  +  6(-3-l/7)  +  2  =  o. 

230.  Simplify: 

I-  (3  +  1/2)'  4.  7(3  +  V^) 

2.  (5  +  V^)'  5-  4(5->^6) 

3-  (7-1/3)'  6.  5(7-1/3)- 

7.  Find  the  sum  of  the  results  of  i  and  4. 

Note  that  6^/2  added  to  6j/2  is  i2]/2.  the  terms  being  similar 
with  respect  to  the  factor  y^2. 

8.  Find  the  simi  of  the  results  of  2  and  5;  of  3  and  6. 
See  the  note  to  problem  7. 


Factoring,  Quadratics,  Radicals  297 

9.  Simplify: 

(1)  {-2-1/2)'  (4)  4(-2-i/2) 

(2)  {-T  +  Vsy  (5)   -9(-7+^8) 

(3)  (S-V^r  (6)  -3(-s-i/^)- 

10.  Find  the  sum — 

(i)  Of  (i)  and  (4)  of  problem  9 

(2)  Of  (2)  and  (5)  of  problem  9 

(3)  Of  (3)  3^^<i  (6)  of  problem  9. 

11.  Solve  and  check  the  results  by  substitution: 

(i)  r^+r  — 7=0  (5)  m'  —  iom=—6 

{2)  d'  +  jd-\-s=o  (6)  v^+3f=+ii 

(3)  x''+4X-i=o  (7)  9^  +  119= +1 

(4)  y^=6y-{-2  (8)  5^  +  205=4-28. 

Exercise  LXVII 

231.  Reduce  to  lowest  terms: 

3a^m  +  2&^m  2ax—6x  —  2ay+6y 


ga^  +  i2a'b'+4b'*  ^akx—gkx—^aky+gky 

2$c^  +  iocd-\-d'  ioklx'*-\-i^klx'  +  2okl 


^ac+ad  +  $bc+bd  2x5  +  2X* + ^x^  +  ^x^  +  ^x  +  4 

232.  Perform  the  indicated  operations: 

5.  -J 1 

^    SX-\-6    x'+4x+4 

x'—xy  x—y 


x'  —  2xy-\-y^     x'+xy 
6       _^     7  ^c- 


io:x;  — 15     4:x;— 6     4Jf^  — i2:x;+9 
4^*+4/  +  i     ^ct  +  ^c+6dt  +  iod 

3^  +  5  2ct+c+4dt  +  2d 

64a'—g6a^b  +  T,^ab'  i6a^  —  i2ab 


4a'—6ab  Sa  —  'j2a^b  +  i62ab'' 


298 


Fir  St- Year  Mathematics 


233.  Multiply: 

I.  (a+b){a-b) 

9- 

2.  ia+y){x-y) 

10. 

3.  (m-n){m-\-n) 

II. 

4.  {r-s){r-\-s) 

12. 

5.  (u+v){u—v) 

13- 

6.  {x^+y){x'-y) 

14. 

7.   (af+:y3)(:>;-)f3) 

15- 

8.  {a'-y'){a^-^y') 

16. 

The  Difference  of  Two  Squares 
Exercise  LXVIII 

{2X+y){2X—y) 
{a  +  5b){a-sb) 

(3W  — 2»)(3W4-2») 

{ab+c)(ab—c) 
(m^n-\-x3){m'n—x3) 
(2U^V  —w){2u'v  +  w) 
(3a^6— 2c4)(3a^6  +  2c4) 
{4abc'  —  3^5)  (4c6c^  +  3^^)  • 

17.  From  these  problems  make  a  rule  for  multiplying  the 
sum  of  two  numbers  by  their  difference. 

1 8.  Make  a  rule  for  factoring  the  difference  of  two  squares. 

Exercise  LXIX 

Factor  the  following  expressions  and  test  the  results,  doing 
all  you  can  mentally: 

14.  4a^cd—2^c^d 

15.  r*—s* 

16.  81W  — i6n4r'* 

17.  kx^—ky^ 

18.  256a4— 625c'*# 


1.  x^'—y 

2.  16^^— 25&» 

3.  4ga'-gb' 

4.  w^n^— i44r* 

5.  289^^—811;' 

6.  i6  —  25y' 

7.  i6gd'h's'  —  22St4 

8.  9w4  — i2in^5"* 

9.  49a^  — ioo6'*c*^ 

10.  196— 36ia'»^'^jf* 

11.  ax*  — 100a 

12.  p^q'r—r 

13.  22^b^°—f*g^h^* 


19.  m^—n^. 

20.  a^°— 6'° 

21.  (r+3.y)='-i6/» 

22.  (2a+&^)^— 9^* 
23-  (5^'-33'^)'-i6z4 

24.  {x'—yY—x^     . 

25.  6^-(3a  +  2c)" 

26.  i6a"  — (2W— 3»)' 


Factoring,  Quadratics,  Radicals  299 

27.  9C»  — (2a  — 3c)"  34.  2^x'-\-i6y'—4a'-h4oxy 

28.  (3a— 26)*  — (2c— 3(/)*  35.  k'—x^  —  2xy—y' 

29.  (4a+5)^  — (2J£;— 3)*  36.  i—a'  —  2ab—b' 

30.  36(a+&)"  — 25(c— <i)^  37.  9m^—a^— 406—46^ 

31.  {a+b+cy-{a-b-cy  38.  36/-=^-4  +  2o/-25P 

32.  a'  —  2ay+b'—c'  39.  :x:^  +  2:x;3'+>'^— a^  — 2a&— &* 

33.  :!c^— 63£;y+9j'  — 162^  40.  a'-\-2a  +  2bc—b'—c'  +  i 

41.  gx'-{-i6y'—4ga'—4b'  +  2Sab  +  24xy 

42.  9a^  — i2a&+4&^  — i6x^— 8xjv— 3'^ 

43.  3[;4  4-5(;2-y2+^4 
Suggestion: 

a;4  +  ar^ys  +  y4  =  JC4  +  23(;')'»  +  ^4 — ic^y*  =  (oca  +  y2 )  2  —  a;2y2 . 

44.  a*  —  'ja'b'+b'^  47.  25:x;'»4-3iic*y2_|_j5^4 

45.  x'^+x'  +  i  48.  a^:x;8+a='j(;4+a* 

46.  i6:x;4  — i7:)(;»j2_^^4        ^g    ^ga^b'i  —  ^^a'b^x' +4X*. 

50.  Give  mentally  the  product:  78*82. 

78-82  =  (8o-2)(8o  +  2)  =80^ -2^  =6,400-4=6,396. 

51.  Give  mentally  the  following  products: 

(i)  19.21  (6)  27.33  (11)  88-92 

(2)  18.22  (7)  26-34  (12)  65-75 

(3)  17-23  (8)  39-41  (13)  87-93 

(4)  29-31  (9)  38-42  (14)  85-95 

(5)  28-32  (10)  67.73  (15)  98-102. 

Exercise  LXX 

234.  Reduce  the  foUoAving  fractions  to  lowest  terms: 

mx^  —  my'  k^m*-\-k^m^n^-\-k^n* 

I. 3.  ■ 

ax—ay-\-bx—by  a'm^+a^mn+a'n' 

x'+ax+bx-\-ab  i2a'*m'+4Sa*  —  iom'b—4Qb 

{x''-a')(x''-b')  ^'  3608-60(146  +  256^ 


300  First-Year  Mathematics 

{Aa-dY-{2h-zcY 
^-  {4a-2by-{3c-dr 


a'  -{-4b'  +gc'  -\-4ab—6ac—i2bc 
i—a'  —  2ab—b' 
^'  i-\-a'+b'  +  2a  +  2b  +  2ab  ' 

235.  In  the  following  exercises  perform  the  indicated  opera- 
tions: 

a^  +  i6a+64  a'  — 16 
a'—Sa  +16  a^—64 
4X'  —  i2xy+gy'     x'-\-a'+b' +  2ax-\-2bx-{-2ab 


x'—a^  —  2ab—b^  4ax'—gay' 

i2a^  — 75&4  ■  240^6—120063  +  150^5 

2abx  +  T,aby  +  2a^b^x  +  T,a'b^y    4X^  +  i2xy+gy' 
x^-\-2abx3-{-a^b^x^  '       mx^+nx^ 

a^—b*  mr-\-ms        mc-\-mb 


a''k'-\-2abk''+b^k^    a'-2ab  +  b'' 

X  X  2X  —  6 

2X—^     2X  +  ;^     4x^  —  9 


2r  — I        4  T,r 

7-  +  ^ 


r  +  i     r^'  —  i 

Exercise  LXXI 
236.  Solve  the  following  equations: 

1.  ak  +  bk=a'—b^.     Solve  for  yfe 

2.  ax-j-^bx=a^+6ab+gb'.     Solve  for  :x; 

3.  w*  — iow  =  75.     Solve  for  w 

4.  a'm—b^m=an-\-bn.     Solve  for  m;  solve  for  » 

5.  r^  =  7/-+44.     Solve  for  r 

6.  T,ut + 6v]^3  -f  ^lut = guh^  -{-2vt  +  i  ^wk^ .     Solvc  f or  t 


Factoring,  Quadratics,  Radicals  301 

7.  m+n-\ — = — .     Solve  tor  :x; 

X        X 
^      X       AX        I         4,4  t^    ,        r 

8.  T — --=1-  -^+-  •     Solve  for  x 
6^     a^      b'    ab    a^ 

Q.  —  =  1 H —  •     Solve  for  m 
8  m 

10.  —  =  --\ h-.     Solve  for?;  for  ^. 

i      g     i      g 

Trinomials  of  the  Form  x''+ax-\-b 
237.  Multiply: 

1.  (x+T,)  by  (^+4)  5.  (x  +  s)  by  (^  +  11) 

2.  (^  +  7)  by  {x  —  2)  6.  (x  +  g)  by  (:x;— 8) 

3.  (x-s)  by  (x+3)  7.  (^-10)  by  (^  +  7) 

4.  (^f— 3)  by  {x—8)  8.  (:x;  — 2)  by  (:x;  — 13). 

9.  From  these  cases  make  a  rule  for  multiplications  of 
this  type. 

10.  The  general  law  thus  obtained  is  proved  as  follows: 

x+a 
x-\-b 


x^-\-ax 

-\-bx+ab 
x'+ax-\-bx-\-ab=x^-\-(a+b)x+ab. 
Therefore  x'  +  (a+b)x+ab  =  {x-\-a){x+b). 

The  sum  of  -\-a  and  +b=a-\-b,  the  coefficient  of  x  in  the 
trinomial.  The  product  of  +a  and  +b  =  -{-ab,  the  term  of 
the  trinomial  not  containing  x  (i.  e.,  the  absolute  term).  This 
gives  us  a  method  of  factoring  by  inspection. 

II.  Factor  :x;^  +  i2X  +  35  by  inspection. 

Find  two  factors  of  +35  whose  sum  is  +12.  The  factors 
are  +5  and  +7.  Then  the  factors  of  x^  +  i2X  +  ;^^  are  (:x;  +  5) 
and  (x  +  y). 


302  First- Year  Mathematics 

12.  Factor  x'—^x—^o  by  inspection. 
The  factors  of  —40  whose  sum  is  —3,  are  —8  and  +5. 
Therefore  x^  —  t,x—4o  =  {x—S){x-\-5). 

Exercise  LXXII 

238.  Factor  by  inspection  the  following  expressions: 

1.  x^  +  ;^x+2  15.  y^  —  i2y—S^ 

2.  m'  +  $m-\-6  16.  d^  —  'jd—^o 

3.  k^  +  i2kb  +  T,sb'  17.  p^—p— go 

4.  j*  +  i4j4-45  18.  r'-iy-4S 

5.  ^^  —  13/^4-40  19.  k'-\-4okb-\-iiib' 

6.  a^+Sa  —  20  20.  x'y'—gxyz  —  ii2z' 

7.  ^=^  +  3^— 180  21.  r=""  — i9r'"+48 

8.  b'  +  jgbc+^SC  22.  k^-yki'-7S 

9.  g^+5'3_42  23.    d^°^  +  2^d5''/^'-^102f^' 

10.  t' -141-51  24.  ife="»-2o)fe'"L%-69L^'" 

11.  r^  — 2r5— 3235"  25.  r^s^/*^  — 1 8r5/3'y2^_ 52^4^2 

12.  v'—6vw—giw'  26.  (w+»)^  +  5(w+»)+6 

13.  a^'ft^— 4a&c='  — 165C4  27.  {a-\-by —  J (a +b)-\- 10 

14.  z'*  — loz  — 24  28.  a^  +  2a6  +  &^  +  3a+36  — 10 

29.  a^—6ab+gb'-\-'jac—2ibc—44c' 

30.  Wx+w^:x;— 65o:x;. 

Quadratic  Equations  Solved  by  Factoring 

239.  Solve  the  equation  ac^— 8:x;+i2=o. 

2C*— 83(;  +  i2=o 
Factoring 

(x— 2)(5£;— 6)=o  (i) 

This  equation  is  satisfied  if 

ac  — 2=0  (2) 

because — 

The  product  of  two  or  more  numbers  is  zero  if,  and  only  if, 
at  least  one  of  the  numbers  is  zero. 


Factoring,  Quadratics,  Radicals  303 

From  equation  (2)  x  =  2. 

Check:  2^  —  8  •  2  +  i2«=4— 10  +  12  =  0. 

Likewise  from  (i)   a:— 6=0, 

Whence,  x  =  6.     Check. 

Consequently,  both  2  and  6  are  roots  of  the  equation. 

1.  Solve   the   same   problem   by   completing   the   square. 
(See  pp.  287,  288.) 

2.  Solve  the  equation  x^=x+42. 

JC*  — .TC— 42=0 

{x—7){x  +  6)=o 

5f  =  7,  or  —6. 

State  the  principle  used  in  the  last  step.     Test  both  results. 

Solve  the  same  problem  by  the  method  of  completing  the 

square. 

Exercise  LXXIII  . 

240.  Solve  the  following  quadratic  equations  by  the  method 
of  factoring,  and  test  results: 

I.  x^—2iX-\-2=o  X'     4^_4, 

X'      X  , 

14.  —=-+9 
02 

4X     X' 

^     5      15    ^ 

16.  -+^=-       , 
2      '      14 

x^     6x 

17.  —  +  —  =  7 

13     13 
x  —  i         X 

ig  5y-i  I  sy-^^-^oy    4 

9  5  9      93' 


i-g    .      4 
4a^     2a  +  3     2a— 3 


2. 

>''-4J+3=o 

3- 

m'=m-\-2 

4. 

w*  +  5«=6 

5- 

a'  +  7a+6=o 

6. 

W"=4W  +  I2 

7- 

k'+k  =  s^ 

8. 

r*  +  5i=2or 

9- 

&»=4&  +  77 

lO. 

C'  +  II2  =  23C 

II. 

^i-^ 

12. 

X^                1T,X 

lO       ^~   lO 

0  +  7 

20.    ^ 

304  First-Year  Mathematics 

Exercise  LXXIV 

241.  Solve  the  following  problems: 

1.  The  base  of  a  triangle  exceeds  the  altitude  by  4  inches, 
and  the  area  is  30  square  inches.  Find  the  base  and 
altitude. 

2.  A  rectangular  field  is  twice  as  long  as  wide.  If  it  were 
20  rods  longer  and  24  rods  wider,  the  area  would  be  doubled. 
What  are  the  dimensions  ? 

3.  The  perimeter  of  a  rectangular  field  is  60  rods.  The 
area  is  200  square  rods.     Find  the  dimensions. 

4.  A  tree  standing  on  level  ground  was  broken  over  so 
that  the  top  touched  the  ground  50  feet  from  the  stump.  The 
stump  was  20  feet  more  than  two-fifths  of  the  height  of  the 
tree.     What  was  the  height  of  the  tree  ? 

5.  A  can  do  a  piece  of  work  in  3  days  less  than  B;  and 
both  can  do  the  work  in  2  days.  How  long  will  it  take  each 
alone  ? 

Forming  Quadratic  Equations  Having  Given  Roots 

242.  By  reversing  the  steps  of  the  solution  of  a  quadratic 
equation,  an  equation  may  be  formed  having  given  roots. 

1.  Examine  again  the  solution  of  the  equation,  x'—8x  + 
12=0,  §239. 

2.  Form  a  quadratic  equation,  whose  roots  are  2  and  6. 

jc— 2  =  0  is  an  equation  having  the  root  2 
x— 6  =  0  is  an  equation  having  the  root  6 

and  (x— 2)(a;— 6)  =0  is  an  equation  having  the  roots  2  and  6. 
Therefore — 

51;'  —  8x  + 1 2  =  o  is  the  required  equation. 

Show  by  substitution  that  2  and  6  are  roots  of  the  last  equation. 

Why  is  the  equation  a  quadratic  equation  ? 

3.  Form  quadratic  equations  having  the  following  pairs  of 
roots: 


Factoring,  Quadratics,  Radicals  305 

(i)       2  and  4  (6)   —J  and  —6 

(2)  2  and  —5  (7)   —8  and  9 

(3)  -3  and  5  (8)       7  and  -9 

(4)  -3  and  -5  (9)       5  and  -12 

(5)  i  and  4  (10)       a  and  6. 

243.  Exercise  (10),  which  gives  the  quadratic  equation 
x'  —  {a+b)x+ab=o,  shows  that — 

The  coefficient  of  the  first  power  of  the  unknown  is  the  sum 
of  the  required  roots,  taken  negatively,  and  the  absolute  term  is 
the  product  of  the  roots. 

1 .  Applying  the  law  just  stated,  form  a  quadratic  equation 
whose  roots  are  2  +  ^/i  and  2  —  y^i . 

The  sum  of  the  roots  is    2  +  ^^2  +  2—^^2=4 
and  the  product  is  (2  +  ^/2)(2—  1/  2)  =4— (]/2)2=4— 2  =  2. 

(See  §229,) 
The  required  quadratic  equation  is  then 

»»  — 4;c+2=o.  (i) 

2.  Show  by  substituting  that  2  +  ^/2  and  2  — y^2  are  roots 
of  equation  (i). 

3.  Form  quadratic  equations  having  the  following  pairs  of 
roots : 

(i)  2  +  1/3    and  2-1/3  (6)   -S  +  Vs    and  -3-^/3 
(2)3  +  1/3    and  3-1/3  (7)   -3  +  1/5    and  -3-1/5 

(3)3  +  1/5    and  3-1/5  (8)   -4  +  1/5    and  -4-1/5 

(4)4+1/7    and4-i/7  (9)   -5  +  1/7    and  -5-1/7 

(5)  5  +  v^ io  and  5  —  1/ 10  (10)   — 6  +  1/2T  and— 6  — v''2i . 

Exercise  LXXV 

244.  Factor  the  following  by  any  method  and  test  results: 

1.  a'bc3  —  2abc3—Sbc3 

2.  6'w+6*n  — 126m  — 126»— 45m— 45« 

3.  p^-26p'  +  2S 

4.  p*  —  i6 


3o6  First- Year  Mathematics 

5.  2ax  +  2ay  —  2az  +  2bx+2by  —  2bz 

6.  v^^—w^^  (into  five  factors) 

7.  in'  —  2mn+n^  —c^  —d'  —2cd 

8.  Sx3-6x'-2Sx  +  2i 

9.  2,k''—6kl  +  T,l^-\-6km—6lm 

10.  ic— 4j  +  i23'z— 92 

11.  (w+n)='-ii(w+«)(c+(i)  +  28(c+rf)» 

12.  afec'^ic  — ioaft<::x;  — 75063? 

13.  Sioa^c^  — 1064^3 

14.  a""  —  ^a"'b^'*  —  ^6b^ 

15.  m^+4n^. 

245.  Reduce  the  following  fractions  to  lowest  terms: 

x'  —  T,x-\-2  2X-\-bx—;^b—6 

x'—4X-\-;^  '  ax—$a+bx—^b 
x'+ax+bx+ab  w*  — 12W  +  32 

X' -\- ;^ax  +  2a'  w^  +  2w  — 24 

2ax-\-;^bx-\-4a+6b  x^"— 4JC"+3 

"^"    :v^+j(;(6  +  2)+2&  ■  x^"-6x"  +  s  ' 

246.  Perform  the  indicated  operations  in  the  following 
problems : 

a^  — 2a— 35     4a3— 9a 
2a3— 3a^         a— 7 

123 


w"  — 7W  +  12     w*— 4W+3     w*— 5m+4 

y—i y+2 y— 3 

y'—yy+io    y"— 9^+14    y"— 12^+35 
/        a'-b'\(  ,  ^a'  +  b'\  .  /a^  ,      .  4&'\ 

3i;'+>''  +  2:j(;y— z'  _  x+y+z 
z'—x'—y'-\-2xy  '  y+z—x  ' 


Factoring,  Quadratics,  Radicals  307 

Trinomials  of  the  Form  ax''-\-hx-{-c 
247.  Multiply: 

I-  (3^  +  5)(2^+3)  5-  (2^-7)(4^+3) 

2.  (5J£:+4)(33f  +  2)  6.  (3^-5)(2^-3) 

3-  (3^-5)(2^+3)  7-  (5^-4)(3^-2) 

4.  (55f+4)(3:v-2)  8.  {2X-']){^x-^). 

9.  Make  a  rule  for  multiplying  binomials  of  this  type. 

The  general  law  thus  obtained  is  proved  as  follows: 

ax  +  h 
cx+d 


OCX'  +  hex 

adx  +  hd 
acx^  +  {bc  +  ad)x  +  bd. 
The  product  of  a  and  c  is  the  coefficient  of  x',  the  product  of  b  and 
d  is  the  absolute  term  (the  term  not  containing  x),  the  sum  of  the  cross 
products  be  and  ad.  that  is,  (be  +  ad)  is  the  coefficient  of  x.     This  gives 
a  method  of  factoring  by  inspection. 

10.  Factor  3:x;*  + 175!; +10. 

Find  factors  of  3  and  factors  of  jo  such  that  the  sum  of  the  cross 
products  is  17. 

The  various  possibilities  are: 

+  3+10  +3v+   ^  +3v+5  +3v+2 

+  i^+   I  +1^  +  10  +1^  +  2  +1^+5 

+  13  cross                +31  cross  +11  cross              +17  cross 

product.               •  product.  product.                product. 

Then  ^x'  +  iyx+io  =  (sx+2){x  + $).     Check. 

11.  Factor  Sy*— 37^2  —  152^. 

Since  the  middle  term  is  negative,  there  are  the  following  possi- 
bilities: 

+    2^-15  +2y+  I  +2y-S  2y-3  +8-15  +8^+  I 

+  4^+   I       +4^-15      +4^+3      4^+5       +i^+    i       +1^-15 
—  58  —26  —14  —2  —7  —119 

+  8+5  +8s,+3 

+  1^-3  +1^-5 

-u)  -37 

Then  8y=  — 37yz—i52»  =  (83^  +  32) (y— 52).     Check. 


3o8  First-Year  Mathematics 

Exercise  LXXVI 

248.  Factor  the  following  expressions,  and  test  the  results: 
T.  2X^-\-iix-\-i2  II.  6&*— 296+35 

2.  8c'+46c  — 12  12.  6/*—/— 77 

3.  3:x;*  — i7:x;  +  io  13.  102  — iia— a* 

4.  8z*— 31Z  +  21  14.  15+372—82' 

5.  5^^—38:^+21  15.  i—()xy+$x^y^ 

6.  iia'—2^ab  +  2b'  16.  2jc*'*  +  ii:x;''  +  i2 

7.  7^^  +  123)^-54  17.   i4:x:""  +  53X'«;y''  +  i4)'''» 

8.  12/^ +315/- 1 55=^  18.  6(a+6)='-7(a+6)-3 

9.  ^m' —  2gmn -{- ^6n^  19.  431;^ +83(;y+ 4^^  +  13^  +  1 3)/ +  3 
10.  lor*  — 23^-5  20.  3c'— 6cJ+3</^  — 2C  +  2J  — 5. 

249.  Factor  the  following  by  any  method  and   test  the 
results: 

1.  2C^xy  —  i;^cdxy+6d'xy  9.  4a^—ga'-\-6a  —  i 

2.  3iif3  +  2a;'— 9^;— 6  10.  a'b^—ji^ab^  —  Tjh^ 

3.  rJC'*  —  2:x;^  + 1  II.  66  +  39w3  +  3m'^ 

4.  6a3— 3oa'6+36a&'  12.  6^"- 246" +  63 

5.  a'—b'—c'  —  2bc-\-a+b+c  13.  0="— a  +  J 

6.  a'—b'+a—b  am      4 

14.  w='-^^+-^ 

7.  i6:x;4-8i  3«     9»' 

8.  a9-256a  i5-  7^'-  07 

16.  a2/"+6v-&25+aV  +  6^/='-a^5. 


Exercise  LXXVII 
250.  Reduce  the  following  fractions  to  lowest  terms: 
2W*+5W  +  2  2X^ -\- ^x -\- ;^ 


I. 


2m='  +  7w+3  '  3:v«  +  7Jc+4 


3- 


Factoring,  Qiuidratics,  Radicals  309 

2a^  +  i'ja  +  2i  i5k'+kl—2i 

3a*  +  26a+35  '  9^*4-3^^— 21 

6d^  —  $d—6  y'z—8yz  +  i$z 


Sd^  —  2d  —  i$  2ay^  —  i^ay  +  2ia 

am'— am— 20a  6x'—bx—i2b 

o. 


2m^  — 7w  — 15  "  gx'—2bx  —  i2b' 

251.  In   the   following   problems  perform    the   indicated 
operations: 

b^'+b-e       b'+Sb  +  is  iobx+^b^_+sx^^(^b+x)x 

2&^  +  5ft  — 3     26^  +  76  —  15  iobx  —  ;^b'—;^x'  '  (:»— 36)6 

3  4  .     y'+y-i       y'-y-i 


i8c^+c— 5    2c^+7c— 4  2y'—y—3    sy'—y—A^ 

y'  —  i4y-\-24  ^    (y'+4y  —  i2)(yw  +  sy) 
w^+9W— 36  '  (w' ■i-2w  —  i$)(yw-{-6w) ' 

252.  Solve  the  following  first  degree  equations  for  the  de- 
sired number. 

1.  2am'  —  ^am-{-2m=bn'+bn—2b.     Solve  for  a;  f or  & 

2.  i^kx^-\-kxy—6ky=gx'-{-^xy  —  2y.     Solve  for  ^ 

3.  tx'—lx  =  2ot-\-2x'—'jx  —  i^.     Solve  for  / 

4.  2a'x+i'jax+2ix—;^a'y—;^^y=o.     Solve  for  ic;  for  j 

5.  mx'  +  i^m-\-i^mx=8mx+2nx'  +  2in.     Solve  for  m; 
for  n. 

253.  Solve  the  quadratic  equation,  20;'  — 13:^+6=0. 

Factoring,  (2^1;— i )(:»;— 6)  =0. 

Then,  by  the  principle  stated  in  §239,  the  equation  is  satisfied  if 
231;— 1=0;  whence  :»  =  J. 

The  equation  is  also  satisfied  if  ac— 6  =  0;   whence  x  =  6. 

The  two  solutions  of  the  quadratic  equation,  2 Ji;'  — 13*+ 6=0,  are 
rc  =  J  and  x  =  6. 

Check  both  results. 


3IO  First-Year  Mathematics 

254.  Solve  the  following  quadratic  equations: 

1.  2:x;^— 53(;  +  2=o 

2.  6j!;"  +  i75(;+i2=o 

3.  6x^=mx-\-7f^m.     Solve  for  m;  for  x 

4.  36^'  =  i9^/+6/.     Solve  for  ife;  for/ 
5-  53'^  +  i3>'=6 

6.  24w^  +  i5n^=38m».     Solve  for  m;  for  » 

7.  2c^+6<i^  =  i3C(i.     Solve  for  c;  for  </ 

8.  8a2  +  i4a6  =  i56^     Solve  for  a;  for  & 

9.  i4:x;=' +  213'^  =  55:^3'.     Solve  for  rx:;  lory 

10.  8^^/^  =  22^/w+2iw^.    Solve  for  kl,  for  k,  and  m 

a*    a     — -? 

II. =— ^ 

7     2      14 


12. 


I  2 


:»— 2     5(;  +  2 


14. 


14         5       3 


3;+4     >'  +  2     y 


I  a;— I  23£;— c 


x  —  i     x^—/\x-\-/^     2:x;^— 6x+4 
255.  Solve  the  following  problems: 

1.  The  length  of  a  rectangle  is  5  greater  than  the  width, 
and  the  area  is  i8|.     Find  the  dimensions. 

2.  B  can  do  a  piece  of  work  in  3  less  days  than  A,  and 
both  can  do  the  work  in  5  j-  days.  How  long  will  it  take  each 
alone  ? 

3.  A  walks  faster  than  B  by  a  quarter  of  a  mile  in  an  hour. 
It  takes  A  a  quarter  of  an  hour  less  than  it  takes  B  to  walk 
15  miles.     Find  the  rate  at  which  each  walks. 

4.  A  freight  train  takes  i|  hours  longer  to  travel  100  miles 
than  a  passenger  train  takes.     The  passenger  train  runs  10 


Factoring,  Qtiadratics,  Radicals  311 

miles  per  hour  less  than  twice  the  rate  of  the  freight   train. 
Find  the  rate  of  each  train. 

5.  In  three  hours  a  boatman  rowed  10  miles  up  a  stream 
and  4  miles  back.  If  the  velocity  of  the  current  was  2  miles 
an  hour  what  was  his  rate  of  rowing  ? 

The  Sum  or  the  Difference  of  Two  Cubes 
256.  Multiply: 

1.  (x-\-y)(x''—xy+y')  4.  (a—b)(a''-{-ab  +  b^) 

2.  (x—y){x^+xy+y')  5.  (a +  26) (0^  —  206+46*) 

3.  (a  +  b)(a''-ab+b')  6.  (sa-b)(ga''-\-:iab-^b') 

7.  (2a  +  36) (4a^— 6^6  +  96=') 

8.  (3a^4- 56')  (904-150^6=' +  2564) 

9.  (7a3  - 46==)  (49^6 +  28a362 +  1664) 

10.  (2a"6"  —  3c")  (4a^"&^" + 6a~6"c~ + gc^*^) . 

11.  Make  a  rule  for  multiplying  the  sum  of  two  numbers 
by  the  square  of  the  first  number  minus  the  product  of  the 
first  and  second  numbers  plus  the  square  of  the  second  number. 

12.  Make  a  rule  for  multipl3dng  the  difference  of  two  num- 
bers by  the  square  of  the  first  number  plus  the  product  of 
the  first  and  second  numbers  plus  the  square  of  the  second 
number. 

13.  Make  a  rule  for  factoring  the  sum  of  two  cubes. 

14.  Make  a  rule  for  factoring  the  difference  of  two  cubes. 

15.  Factor  6403 +  27 &3. 

64a'  +  276^  =  (4a  +  3&)  (i6a'  —  1 2ab  +  gb'). 
The  expression  is  the  sum  of  two  cubes  (40)3  and  {sb)3.  Therefore 
one  factor  is  the  sum  of  the  cubes  of  the  numbers  (4a  and  3J).  The 
other  factor  is  the  square  of  the  first  number  of  the  first  factor,  (40)^  = 
i6a»,  minus  the  product  of  the  first  and  second  numbers,  —(^aX^b)  = 
—  i2ab,  plus  the  square  of  the  second  number  (3&)2=96».  Check  the 
result. 

16.  Factor  83£;3 -125)^3. 

Sx'  —  i2^y'  =  {2X—  5y){4x'  +  ioxy  +  2Sy^).     Check  the  result. 


312  First- Year  Mathematics 

257.  Factor  the  following  expressions,  doing  as  many  a" 


can  mentally: 

I.  a^+h^ 

II. 

^12C^  —  2']d^ 

2.  %x3-y3 

12. 

^3/3+343 

3.    W3  +  27n3 

13- 

a3'»4-ft3'« 

4.  9>c3-d^ 

14. 

8;c3''-y3'- 

5-  343-^^ 

15- 

2ym3''+8n(''' 

6.  /3-f-64 

16. 

{a+by+c3 

7.  8:^18  +  273" 

17- 

{a+my—n^ 

8.    2  7M97t)6_j_j 

18. 

a3  +  (fe+c)3 

9.  8z'^*  +  27w'8 

19. 

(w+3)3-a3 

10.  729a^  +  2i6c'^ 

20. 

(5m— «)3+C 

The  Remainder  Theorem 

1.  What  is  the  remainder  after  dividing  3X  +  7  by  rx;  — i  ? 
What  is  the  value  of  3:x;4-7  if  i  is  substituted  for  x  ? 

2.  What  is  the  remainder  after  dividing  4X-i-^  by  x  —  2? 
What  is  the  value  of  4:x;+5  if  2  is  substituted  for  :x;  ? 

3.  What  is  the  remainder  after  dividing  3:x;— 6  by  x  —  2? 
What  is  the  value  of  3:^;— 6,  if  x  =  2? 

258.  In  the  following  problems  give  the  remainders  after 
divid  ng  and  evaluate  as  required: 

1.  Divide  4:^—3  by  :x;— 3.     Evaluate  the  dividend  for  :»  =  3. 

2.  Divide  iix  — 13  by  x— 10.     Evaluate  the  dividend  for 
x  =  io. 

3.  Divide    2X+3    by   x-\-i.     Evaluate    the   dividend    for 
x=—i. 

4.  Divide  ii:x;  +  25  by  x-\-2.     Evaluate  the  dividend  for 

X=—2. 

5.  Divide  2o:x;— 37  by  :x;  +  5.     Evaluate  the  dividend  for 

6.  Divide  ax+b  by  x—r.    Evaluate  the  dividend  for  x=r. 


Factoring,  Qtmdratics,  Radicals  313 

7.  Divide  my—n  by  y—c.    Evaluate  the  dividend  for  y=c. 

8.  Divide  ax-\-hhy  x-\-r.    Evaluate  the  dividend  for  x=—r. 

9.  Divide  mv+n  by  v-{-t.   Evaluate  the  dividend  for  v=  —t. 

10.  Divide  x3-\-^x^—6x  —  ii  by  x  —  2.  Evaluate  the  divi- 
dend ioT  X  =  2. 

11.  When  an  expression  in  x  is  to  be  divided  by  x  plus  a 
number,  or  x  minus  a  number,  how  may  the  remainder  be 
obtained  without  dividing  ? 

In  the  following  problems  give  the  remainders  without 
dividing  and  check  the  results  by  division: 

I-  (3^-7)-^(^-i)  4-  {x^+ii)-i-(x+7,) 

2.  (7x+s)^{x  +  i)  5.  (x^  +  sx-s6)^(x-4) 

3.  {^X-\-Il)-7-{x-\-2)  6.    {2Xi-{-X^—^X  +  'j)-7-{x+S). 

7.  Divide  a:x;^ +6.^-1- c  by  :x;—r. 

x  —  r)ax^+bx  +  c{ax  +  b  +  ar 
ax^  —  axr 

bx  +  axr 
bx—br 


axr  +  hr  +  c 
axr  —  ar^ 


Remainder,  ar^  +br  +  c. 
Obtain  the  remainder  by  the  rule  given  in  problem  1 1 . 

8.  Without  dividing,  give  the  remainder  after  dividing 
kx^+lx^+mx+n  by  x  —  i.     Check  by  division, 

9.  Without  dividing,  give  the  remainder  after  dividing 
x^ +px+q  hy  x-\-k.     Check  by  division. 

259.  If  in  a  division  problem  the  remainder  is  zero,  the 
division  is  exact,  and  the  divisor  is  a  factor  of  the  dividend. 

260.  Determine,  without  dividing,  whether  the  first  ex- 
pression is  a  factor  of  the  second  in  the  following,  and  factor 
the  factorable  expressions: 

1.  x  —  2 3^—6 

2.  X  —  l: X^—^X  +  2 


314  First-Year  Mathematics 

3.  x-^\ 5(;^  — I2X  — 13 

4.  x  —  2 x^—x^-\-ix-\-2 

5.  x  +  'j :!c3  +  5x*  — 29X  — 105. 

261.  The  proof  will  now  be  given  of  the  theorem  we  have 
been  using,  i.  e.,  that,  if  in  a  given  expression  containing  x 
(call  it  for  brevity  an  expression  in  :x;),  r  is  put  for  x,  the  ex- 
pression in  X  reduces  to  the  remainder  R,  that  is  obtained  by 
dividing  the  expression  \n  xhy  x—r. 

In  division  there  is  always  a  divisor  d,  a  dividend  D,  a 
quotient  Q,  and  a  remainder  R  {somstim^s  zero),  and  the  fol- 
lowing relation  connects  them: 

D=QXd+R. 

Then,  if  the  expression  in  x  is  divided  by  x—r,  by  this 
principle  we  have  the 

expression  in  x=Q(x—r)-\-R. 

Substitute  on  both  sides,  for  x,  the  number  r,  then  the 
expression  in  r=Q(r—r)-{-R  =  QXo-\-R=R. 

Therefore,  R,  the  remainder  found  by  dividing  the  expression 
in  X  by  x—r,  equals  the  expression  in  r,  i.  e.,  the  given  expres- 
sion with  r  put  in  place  of  x. 

This  is  called  the  remainder  theorem. 

262.  Multiply  the  following  expressions  and  make  a  rule 
for  finding  the  constant  term  in  the  product  from  the  constant 
terms  of  the  factors: 

{x-\-4){x-\-s)  6.  {x-5){2X^-\-x-8) 

(x-3)(:x;  +  7)  7.  (x-\-2){x-\-s){x-\-4) 


{x-2){x-6)  8.  (3C  +  i)(^  +  3)(^-5) 

{x-\-2){x'+S)  9.    (x  +  2){x-s)(x-2) 


{x-\r3)ix'+x-4)  10.  (^-3)(:x;-4)(^-5)- 

II.  From  problems  i-io  it  is  clear  that  if  an  expression 
with  integral  coefficients  has  a  factor  x—a,  then  a  is  a  factor 
of  the  constant  or  absolute  term  of  the  ejJpression. 


Factoring,  Quadratics,  Radicals  315 

12.  Factor  by  the  remainder  theorem,  x^  —  ']x'^-\-i6x  —  i2. 

The  exact  divisors  of  —12  are  i,  —  i,  2,  —2,  3,  —3,  4,  —4,  6,  —6 
and  12,  —12. 

For  x—i,  the  value  of  xi~']x^->ri6x—  12  is  —2. 

For  a;=  —  I,  the  value  is  —36 

For  x  =  2,  the  value  is  o. 

Therefore,  x—  2  is  a  factor  of  xi  —  ^x^  +  i6:k—  12. 

By  division,  the  other  factor  is  x''  —  ^x->r  6. 

Then,  x3  —  'jx^  +  i6x—i2  =  {x—  2){x—  2){x—7,). 

263.  Factor  by  the  remainder  theorem: 

1.  5!;3_i  6.  3'3-|-64 

2.  :x;3  — 7a:*  +  7x  +  i5  7.  r3+r*  — 22^—40 

3.  ic^— 5:x;^  — 2:x;  +  24  8.  2w3+gm^  +  i2w+4 

4.  a;3+8  9.  :x;4— 32c3  — 2irc*+43^+6o 

5.  x^-\-2ax^-\-<,a^x-\-^a^  10.  x-*  — i55(;'  +  io:x;  +  24. 

264.  Solve  x^  +  3:x;^  — 13:)£;  — 15=0. 

«=  — I,  reduces  the  expression  on  the  left  to  zero,  then  «— (— i), 
or  a;  + 1  is  a  factor. 

By  dividing  we  get: 

af3 +  3J(;2— 13:11;— 15  =  (a;+ l)(ai;2  +  2JC— 15) 
=  (^+i)0>.-3)(«  +  S). 
Then,  {x-\-i){x-t,){x-\-^)=o, 
Whence,  r!c=  —  I,  3,  —5.     Why?     Check. 

265.  Solve  the  following  equations: 

1.  w3— 6m^  +  iiw— 6  =  0     4.  x^  +  <,x'—/^i,=gx 

2.  5(;3  +  2:x;^  =  ii:x;  +  i2  5.  y^-\-i2o='jy^  +  i/\.y 

3.  4:x;  +  i6=x3+4:x;^  6.  k^-\-2k''—^k—2>=o 

7.  3(;3=5a;»  — ia»+i2 

8.  4>''' +32^3 -|-833'»  +  76;y  + 21=0 

9.  a'*  —  iia3+44a*  — 760+48=0 

^*  .    1,5         I     ,  ,  1,4       4^-1 


12       20;^       I2A;      2X'  X'      4iC*  +  I2 


31 6  First-Year  MatJumatics 

266.  Show  by  the  remainder  theorem  that  the  following 
have  a— 6  as  a  factor  and  find  the  other  factor. 

1.  a*— 6*  5.  a^—b^ 

2.  a3_ft3  6.  a^'-b^ 

3.  a*—h*  7.  a*— 6* 

4.  as-fts  8.  a^-h^. 

9.  Make  a  rule  for  forming  the  other  factor  in  1-8. 

267.  Show  by  the  remainder  theorem  that  the  following 
have  a +6  as  a  factor,  and  find  the  other  factor. 

1.  a^^-hi  6.  «*-&== 

2.  flS -1-^,5  7.    a4-64 

3.  a7+67  8.  a^-h^ 

4.  a^+ft'  9.  a^—b^ 

5.  a" +6"  10.  a'°-6^°. 

11.  Make  a  rule  for  forming  the  quotient  of  the  sum  of 
the  same  odd  powers  of  two  numbers  by  the  sum  of  those 
numbers. 

12.  Make  a  rule  for  forming  the  quotient  of  the  difference 
of  the  same  even  powers  of  two  numbers  by  the  sum  of  those 
numbers. 

Miscellaneous  Exercises 

268.  Factor,  and  check  the  results: 

1.  m'—n'  8.  450  — 2a^ 

2.  4a='— 96^  9.  g^+h^ 

3.  I  —  r^  10.  a^-{-b^  (3  factors) 

4.  x'y—y3  II.  ac^+j" 

5.  u3+v3  12,  m^'—n" 

6.  c^—d^  13.  w^-\-4 

7.  as+fcs  14.  a'^b^—Ga^b  +  ga' 


Factoring,  Quadratics,  Radicals 


317 


15.  x^y^  +  2X^y^z^  +y'z^ 

16.  I3mn''  —l'mn'x—l''m'n' 

17.  y'— ay— 42a' 

18.  56^  +  io&-is 

19.  T,a^—gab  —  2iob^ 

20.  gm^ —  24mn  +  i6n^ 

21.  4J£;^+32:)(;>'+39J^ 

22.  6cj+acjv^+&j3-f  aj4 

23.  m^—6bm—4+gb' 

24.  72^^+41^-45 

25.  3c3-|-;x;^— 3:x;  — 2 

26.  6a^«+a"6"-i56^'' 

27.  a46^+a363_o3ft2c 


29.  a4  4-26a^6»  +  256'» 

30.  ^oc'—cd  —  2od' 

31.  27+a366c9 

32.  aS  +  1024 

33.  io:x;V+33^r-7r 

34.  a*+6^— c^  — 2a6 

35.  a^— 6=^+0  +  6 

36.  no— X— iC^ 

37.  4:x;'*— 6i:!C^;y^  +  9v'» 

38.  ac— aic— 76c+7&3(; 

39.  3w3+5w^  — 7w  — I 

40.  af*  — i3:x;^+36 

41.  24:x;^  +  373(;3'  — 72;y^ 

42.  6{a-\-by  —  {a+b)x  —  i2x^ 


28.  ioam+45a^fe^  +  2564 

43.  esa^'c'+Sacd—d' 

44.  w"+6w— x^  +  9— 4:x;j— 4;y* 

45.  i2j3-|-3^2— 8;y— 2  47.  a^+b^+a+b 

46.  3x3  +  2:x;^-2:x;-i  48.  4a='6*  — (a^+fe^-c^")* 

49.  9?w^— 9a^+64w^  — 646^+48ww+48a6 

50.  x^-{-gy^  +  2^z^—6xy  —  ioxz+;^oyz. 

269.  Solve  the  following  equations,  and  check  results 

1.  ay-^by—cm=ac+bc—my.     Solve  for  ;y 

2.  2a^r  —  ioa3b  +  55ab'c'  =  iibc'r.     Solve  for  & 

3.  ^^  +  240^  =  14a/.     Solve  for  t;  for  a 

4.  m^=4m  +  i.     Solve  for  w 

5.  i6:x;  +  i5  =  i5a(;'.     Solve  for  :x; 

6.  «3-f  6o=4w^  +  i7w.     Solve  for  » 

7.  aj  =  fe>'+r+^.     Solve  for  ;y 

8.  3^  =  5^j  +  3<3^  —  S'l}'  +  3^  —  Sby-    Solve  for  ife 


31^ 


First-Year  Mathematics 


9 

lO 

II 

12 

13 
14 

15 
i6 

17 
i8 
19 
20 


V' 4-3^  +  7=0.     Solve  for  v 
w'+8=o.     Solve  for  w 

X 


2X^  X 

X^—1      X  —  l      x-\-i 


+3.     Solve  for  x 


2X  A  16  X 

1 — =—. and  — v=o.     Find  x  and  y 

x-T,y    X    x{x-2,y)  3     ' 

2  II 

— 1-»=4,     and 1 =2.     Find  w  and  « 

m  m  —  i     n  —  2 

a       b       c 

1 i — =d.     Find  ar 

mx    nx    rx 

ax         b 

r =--a^+b' 

a'—b^     nx 

bx      3C+36 


a—b     a+b    * 

5w^  +  7w+4_3w*+6w+7 

^  +  3     Jg— 3_2JC— 3 

X+2      X—2        X—I 

f-^ =4 

x'  +  2      2:x;  — I 

X  2  X 

--\ =  2 . 

2       X^-{-X  2 


270.  Reduce  to  lowest  terms: 
x3—4x'+x-\-6 
af3— 63(;»  +  iia;— 6 
ac—bc—ad  +  bd 


2. 


ac-\-ad—bc—bd 


A- 


2W3  +  5W^— I2W 
7W3+25W^  — I2W 

a'  —  2ab-\-b''—c'' 
b'-2bc+c''-a' 


x^  —  ^x'y  —  2xy'  +  243/3 
x^+^x^y  —  ioxy'  —  24y^ 

271.  Perform  the  indicated  operations: 

x+6    x+4.  ( 


ac— 3     51;  — 2 


3a— 36     2b  — 2a 


Factoring,  Quadratics,  Radicals  319 

m^—Q         :x;^+^— 20 


m^+m  — 12       x^  —  2$ 

y+s    y—4    y'+y— 20 
r'+r—s  r^'+r  —  i 


2r'—iir+i2     2r^  +  5r— 12 
a'"         a'"  I  I 

6.   ■ ; h 


a"  — I     a"  +  i    a"  — I     a"  +  i 

7.    =^  •  =^   •  ~ 

x3—y3     xi-\-y3     x^'+y^ 

3g'— 4JC+3     x^+Sx  +  i^  _  Jg'  +  2X— 15 

iC^  +  5^  +  6        iC^— 5:^+4     *    X^— 2X  — 8 

/     ,  r/\/      m^\  ah  (h     a\ 

\a— :x;      /     \      a+xf 
y'  —  2ay+a^      y'  —  ga'        y^+Say 
y^-{-4ay—$a'    ay  +  2a'    y'—^ay+^a 

I     ,       r  +  x       ,        r—2 
14.  - — +-; — -V-+- 


f— I     r^— 3r4-2     2r='— 5^  +  2 

gS— 65  a  +  6  a4-&4       ga+fca 

^^'    a-b   '  {a-by  '  {a  +  b)3~a'-b' ' 

272.  Simplify: 

X    y 

2~6 

y 

3.-- 

Multiply   the   numerator   and   denominator   by   30,    the   common 

X    y  y 

denominator  of  the  fractions  -,  -z,  and  -,  whence  the  complex  fraction 
265'  ^ 

reduces  to  ^f^~P  =  i . 
i8x—6y 


320  Fir  si- Year  Mathematics 

273.  Simplify: 

a    a'    a^ 


9. 
a 

x+- 


x-5 


y- 

-2 

y 

r 

—3 

!  —  ] 

\2    3'*4-6j4-9 

6y 

^  +  7^  +  2 

r 

—  V  — 12 

m 

'  + 

n' 

n 

m^— w' 

x—2  II  m3+n3 

,6  X    a  X    a 

ft-i+---  -+ — 2     -+-+2 

0  —  6  6.  a     :x;  a     :j; 

3-  —  „     „    + 


6-2+- 


rv+a 


6-6 

274.  Solve  the  following: 

1.  One  farmer  can  plow  a  field  in  r  days,  and  another  can 
plow  it  in  s  days.  In  how  many  days  can  they  plow  it,  both 
working  together  ? 

2.  A  can  do  a  piece  of  work  in/  days,  B  in  ^  days,  C  in 
h  days.  How  many  days  will  it  take  all  together  to  do  the 
work? 

3.  The  height  of  a  bin  is  2  feet  more  than  the  width  and 
one  foot  more  than  the  length  and  the  capacity  is  336  cubic 
feet.     Find  the  dimensions. 

4.  The  ratio  of  the  capacities  of  two  bins  is  a:  6,  and 
together  they  hold  a^+fe^  bushels.  What  is  the  capacity  of 
each? 

5.  A  has  m  times  as  much  money  as  B.  Their  total  capital 
is  invested  in  the  following  amounts:  (/+^)  dollars,  (m  +  2^) 
dollars,  and  {2k+km)  dollars.     How  much  money  has  each? 

6.  A  quantity  of  flour  was  bought  for  $96 .00.  If  8  barrels 
more  had  been  obtained  for  the  same  amount  of  money,  the 


Factoring,  Quadratics,  Radicals  321 

price  per  barrel  would  have  been  $2.00  less.     How  many 
barrels  were  bought  ? 

7.  A  and  B  are  traveUng  in  the  same  direction,  and  B  is 
k  yards  in  advance  of  A.  If  A  travels  at  the  rate  of  n  yards 
per  minute,  and  B  at  the  rate  of  m  yards  per  minute,  in  how 
many  minutes  will  A  overtake  B  ? 

8.  N  equal  cubical  boxes  with  edges  equal  to  w+3  inches 
have  a  total  capacity  of  648  cubic  inches.  Find  the  number 
of  boxes. 

Summary 

The  factors  of  a  number  are  the  exact  divisors  of  the  number . 

A  prime  number  is  a  number  that  has  no  factors  except 
itself  and  unity. 

A  prime  factor  is  a  factor  that  is  a  prime  number. 

An  expression  is  equal  to  the  product  of  all  its  prim^  factors 
for  all  values  of  its  letters.     Such  equality  is  called  identity. 

An  expression  in  which  the  highest  power  of  the  unknown 
is  the  second  power  is  a  second  degree  expression,  or  a  quad- 
ratic expression. 

An  equation  in  which  the  highest  power  of  the  unknown 
is  the  second  power  is  a  second  degree  equation,  or  a  quadratic 
equation.  A  quadratic  equation  is  sometimes  called  a  quad- 
ratic. 

An  expression  of  the  form  ab-{-ac  is  factored  thus: 
ab-Yac=aib-\-c). 

An  expression  of  the  form  am-\-bm+an+bn  is  factored  thus: 
am  +  bm-\-an  +  bn  =  {m+n){a+b). 

An  expression  of  the  form  x^+mx+n  is  factored  thus: 
x'  +  {a-\-b)x+ab  =  (x+a){x-\-b) ,  where  m=a+b  ,  and  n=ab. 

An  expression  of  the  form  x^—y'  is  factored  thus: 
x''—y''  =  (x+y)(x—y). 


322  First-Year  Mathematics 

An  expression  of  the  form  ax'  +bx+c  is  factored  by  finding 
a  pair  of  factors  of  ax'  and  a  pair  of  factors  of  c  the  sum  of 
whose  cross  products  is  bx. 

Numbers,  such  as  1/3, 1/5, 1/7,  etc.,  that  can  be  expressed 
only  approximately  without  the  use  of  the  radical  sign  are 
called  irrational  numbers.  When  expressed  with  the  aid  of 
the  radical  sign,  irrational  numbers  are  called  radicals. 

The  product  of  two  or  more  numbers  is  zero,  if,  and  only  if, 
at  least  one  of  the  numbers  is  zero. 

In  division  the  dividend,  D,  the  quotient,  Q,  the  divisor,  d, 
and  the  remainder,  R,  are  connected  by  the  relation 
D=Q  '  d+R. 

The  remainder,  R,  foxmd  by  dividing  an  expression  in  x 
by  x—r,  equals  the  expression  in  r,  i.  e.,  the  given  expression 
with  r  put  in  place  of  x.     This  is  called  the  remainder  theorem. 


CHAPTER  XIV 


POLYGONS.     CONGRUENT   TRIANGLES.     RADICALS 
Interior  Angles  of  Polygons 

275.  A  plane  figure  bounded  by  straight  lines  is  a  polygon. 

1 .  Find  the  sum  of  the  interior  angles  x,  y,  z,  t,  w,  of  a  five- 
sided  polygon  (Fig.  242). 

From  a  point  P,  inside 
the  figure,  draw  lines  to  the 
vertices  (Fig.  243). 

How  many  triangles  are 
thus  formed  ? 

What  is  the  sum  of  the 
angles  of  each  triangle  ?  Of 
all  the  triangles? 

What  is  the  sum  of  the  angles  around  P  ? 

Prove  that  t  +  w  +  x  +  y  +  z  =  $  •  180—360. 

2.  Prove  that  the  sum  of  the  interior  angles 

(i)  of  a  6-sided  polygon  is  6  •  180—360 

(2)  of  a  7-sided  polygon  is  7  •  180—360 

(3)  of  a  lo-sided  polygon  is  10  •  180—360 

(4)  of  an  5-sided  polygon  is  5  •  180—360 

(5)  of  an  «-sided  polygon  is  n  •  180—360. 

276.  A  diagonal  of  a  polygon   is  a  straight  line  joining 
two  vertices  that  are  not  consecutive. 


Fig.  242 


Fig.  243 


Convex  Polygon       Concave  Polygon 

A  polygon  of  three  sides  is  a  triangle. 
A  polygon  of  four  sides  is  a  quadrilateral. 

323 


324 


First-  Year  Mathematics 


A  polygon  of  five  sides  is  a  pentagon. 

A  polygon  of  six  sides  is  a  hexagon. 

A  polygon  of  seven  sides  is  a  heptagon. 

A  polygon  of  eight  sides  is  an  octagon. 

A  polygon  of  ten  sides  is  a  decagon. 

A  polygon  of  fifteen  sides  is  a  pentedecagon. 

A  polygon  of  n  sides  is  an  «-gon. 

277.  If  all  sides  are  equal,  the  polygon  is  equilateral. 


If 


Equilateral  Quadrilateral     Equilateral  Octagon 
all  interior  angles  are  equal,  the  polygon  is  equiangular.     A 


Equiangular  Hexigon     Equiangular  Quadrilateral 

polygon  that  is  both  equilateral  and  equiangular  is  a  regular 
polygon. 


Regular  Hexagon  Regular  Quadrilateral 

278.  Theorem:  If  d  represents  the  number  of  degrees  in 
the  sum  of  the  interior  angles  of  a  polygon,  and  n  the  number 
of  sides,  then 

d=n  •  180  —  360  (see  §275).  (i) 


Polygons,  Congrtient  Triangles,  Radicals       -      325 

1.  Show  that  equation  (i)  may  be  written 

J  =  (w -2)180.  (2) 

2.  Letting  r  stand  for  the  number  of  right  angles  in  the 
sum  of  the  interior  angles  of  a  polygon  of  n  sides,  show  that 
equation  (2)  may  be  written 

r  =  (»-2)2.  (3) 

3.  Find  from  equations  (2)  and  (3)  the  ratio,  d:  r. 

4.  Using  equation  (2)  as  a  formula,  find  the  sum  of  the 
interior  angles  of  a  quadrilateral;  of  a  pentagon;  of  a  hexa- 
gon;  of  a  triangle. 

5.  Using  equation  (3)  as  a  formula,  find  the  sum  of  the 
interior  angles  of  a  heptagon;  of  an  octagon;  of  a  decagon; 
of  a  pentedecagon ;  of  a  polygon  of  18  sides. 

6.  In  the  equation  <f  =  (w  — 2)180: 

(i)  What  is  the  value  of  cJ,  if  «  is  3  ?  9  ?  16  ?  5  ? 
(2)  What  is  the  value  of  n,  if  d  is  900  ?  7200  ? 

7.  Solving  for  n  the  equation  </  =  (»  — 2)180, 

Show  that  «=^i^.  (4) 

180  ^  ' 

8.  By  means  of  formula  (4),  find  the  number  of  sides  of  a 
polygon,  the  sum  of  whose  interior  angles  is  3600°;  17100°; 
180°. 

9.  Solving  for  n  the  equation  r  =  {n  — 2)2, 

Show  that  »= .  (5) 

10.  By  means  of  formula  (5),  find  the  number  of  sides  of 
a  polygon,  the  sum  of  whose  interior  angles  is  80  right  angles; 
192  right  angles;   2  right  angles. 


326 


First- Year  Mathmatics 


11.  Each  interior  angle  of  a  regular  pentagon  is  x.  Find 
X  in  degrees. 

In  formula  (2),  p.  325,  let  «  =  s;  thend  =  5a;  (why?),  and  5*  =  (5  — 2) 
180.     Solve  for  x. 

12.  Find  the  number  of  degrees  in  each  interior  angle  of 
a  regular  hexagon;  of  a  regular  octagon;  of  a  regular  17-gon; 
of  a  regular  w-gon. 

13.  Using  the  equation  obtained  in  the  last  part  of  problem 
12  as  a  formula,  find  the  number  of  degrees  in  each  interior 
angle  of  a  regular  heptagon;  of  a  regular  decagon;  of  a  regu- 
lar pentedecagon. 

14.  An  interior  angle  of  a  regular  polygon  is  120°.  How 
many  sides  has  the  polygon  ? 

In  formula  (2),  p.  325,  J  =  i2on.     Why? 
Then  I2om  =  (»—  2)180.     Solve  for  n. 

15.  How  many  sides  has  a  regular  polygon,  one  of  whose 
interior  angles  is  135°?    140°?    144°?    170°? 

16.  Can  a  tile  floor  be  laid  with  tiles  all  having  the  shape 
of  regular  polygons  of  3  sides?  Of  4  sides?  Of  5  sides? 
Of  6  sides  ?    Of  8  sides  ?    Of  1 5  sides  ? 

17.  In  Fig.  244,  show  that  x=x'. 


Fig.  244 


Fig.  245 


18.  In  Fig.  245,  a=a',  b=b'.     Show  that  c=c\ 

19.  In  Fig.  246,  a=a',  b=b'.     Show  that  c=c\ 


Polygons,  Congrtcent  Triangles,  Radicals 


327 


20.  In  triangle  ABC,  Fig.  247,  A  O  is  perpendicular  to 
B  C,  and  ZB  =  ZC.     Show  that  y=y'. 


Fig.  246 


Fig.  247 


21.  In  Fig.  248,  w=w',  x=x'.  Show  that  A  O  is  perpen- 
dicular to  B  C. 

If  two  angles  of  one  triangle  are  equal  to  two  angles  of  another, 
the  third  angle  of  the  first  triangle  is  equal  to  the  third  angle 
of  the  second. 

Exterior  Angles  of  Polygons 

279.  If  one  side  of  an  «-gon  is     - — ^^_ 
prolonged  at  each  vertex,  n  exterior 
angles  are  formed,  taking  one  at  each 
vertex;  for  example,  angles  a,  b,  c,  d, 
«>  /  (Fig-  249)- 

I.  Find   the   sum  of  the  exterior 
angles  of  an  »-gon. 

The  sum  of  the  interior  angle  and  the 
exterior  angle  at  each  vertex  is  180°. 


Fig.  249 


In  Fig.  249,     0+^  =  180,  b  +  h  =  180,  c  +  k  =  i8o,  etc. 

Since  there  are  «  vertices,  the  sum  of  all  the  angles,  interior  and 
exterior,  is  n  times  180  degrees,  or  i8on  degrees. 

i8on  is  the  sum  of  all  interior  and  exterior  angles 

i8on— 360  is  the  sum  of  all  interior  angles — §278. 


Subtracting,  360  is  the  sum  of  all  exterior  angles. 


328  First-Year  Mathematics 

2.  Rotate  a  pencil  through  all  the  exterior  angles  of  a 
polygon,  taking  one  at  each  vertex.  Through  what  part  of  a 
complete  turn  has  the  pencil  rotated  ? 

3.  The  sum  of  the  interior  angles  of  a  polygon  is  7  times 
the  sum  of  the  exterior  angles.     Find  the  number  of  sides. 

Show  that  (n— 2)180  =  7  'S'^o.     Solve  for  n. 

4.  How  many  sides  has  a  polygon  in  which  the  sum  of  the 
interior  angles  is  9  times  the  sum  of  the  exterior  angles  ? 

5.  How  many  sides  has  a  polygon  in  which  the  sum  of  the 
interior  angles  is  a  times  the  sum  of  the  exterior  angles  ? 

6.  Each  exterior  angle  of  a  regular  polygon  is  15°.  Find 
the  number  of  sides. 

7.  Each  exterior  angle  of  a  regular  8-gon  is  x.  Find  x 
in  degrees. 

8.  Each  exterior  angle  of  a  regular  «-gon  is  y.  Find  y 
in  degrees. 

9.  An  exterior  angle  of  a  regular  polygon  is  \  of  the  adja- 
cent angle.     Find  the  number  of  sides. 

280.  Some  of  the  following  definitions  have  been  stated  in 
preceding  chapters;  they  are  here  given  again  for  convenience. 

TRIANGLES 

281.  An  isosceles  triangle  is  a  triangle  that  has  two  equal 

sides. 


Isosceles  Triangle  Scalene  Triangle 

282.  A  scalene  triangle  is  a  triangle  that  has  no  two  equal 

sides. 


Polygons,  Congruent  Triangles,  Radicals 


329 


283.  A  right  triangle  is  a  triangle  that  has  a  right  angle. 
The  side  opposite  the  right  angle  is  the 

hypotenuse. 

The   symbol   for  triangle  is  A  ;    for  tri- 
angles is  A- 

QUADRILATERALS 

284.  A  trapezoid  is  a  quadrilateral  having      Right  Triangle 
one  pair  of  parallel  sides. 


IZ7  [z:KX 


Parallelogram         Trapezoid        Rhombus        Rectangle    Square 

285.  A  parallelogram  is  a  quadrilateral  having  iwo  pairs 
of  parallel  sides. 

286.  A  rhombus  is  an  equilateral  quadrilateral, 

287.  A  rectangle  is  an  equiangular  quadrilateral. 

288.  A  square  is  a  regular  quadrilateral. 
Show  that  a  parallelogram  is  a  trapezoid. 
Show  that  a  square  is  a  rectangle. 
Show  that  a  square  is  a  rhombus. 
Is  every  trapezoid  a  parallelogram  ? 
Is  every  rectangle  a  square  ? 
Is  every  rhombus  a  square  ? 


Congruent  Triangles 

289.  Inaccessible  distances  frequently  may  be  determined 
by  triangles  that  are  both  equal  and  similar. 

I.  To  find  the  distance  from  a  point  A  to  a  point  B  on 
the  opposite  side  of  a  river  (Fig.  250)  a  surveyor  lays  off 
line  C  D  making  angle  x  equal  to  the  angle  x',  and  the  line 
A  D,  making  angle  y  equal  to  the  angle  y'. 


330 


First-Year  Mathematics 


What  corresponding  parts  of  triangles  ABC  and  ADC 
are  equal  ? 


Fig.  250 

How  do  triangles  ABC  and  ADC  seem  to  compare  as 
to  size  and  shape  ? 

A  D  is  measured  and  found  to  be  310  feet  long.  What 
seems  to  be  the  length  of  A  B  ? 


2.  Draw  on  paper  a  triangle  as  A  B  C  (Fig.  251).  Con- 
struct another  triangle  A'B'C  as  follows: 

make  A'B'=AB 
ZA'=ZA 
ZB'=ZB. 

Use  the  protractor,  or  the  construction  given  in  problem  24,  pp.  148,  149. 

Cut  triangle  A'B'C  from  the  paper  and  see  if  it  can  be 
made  to  fit  upon  triangle  ABC. 

How  do  the  triangles  ABC  and  A'B'C  compare  as  to 
size  and  shape  ? 

Triangles  which  can  be  made  to  fit  one  over  the  other,  or 
to  coincide,  are  called  congruent  triangles. 


Folygons,  Congruent  Triangles,  Radicals  331 

If  A'B'C  can  be  made  to  fit  on  A  B  C,  what  must  be 
true  of  the  relations  of  the  parts  B  C  and  WC  ?  Of  the  parts 
C  A  and  C'A'  ? 

3.  Draw  the  triangles  of  problem  2  on  the  blackboard. 
On  thin  paper  make  a  trace  of  A'B'C  with  colored  crayon, 
and  fit  the  trace  over  ABC. 

In  congruent  triangles  the  parts  (the  angles  and  the  sides) 
which  coincide,  when  the  figures  are  made  to  coincide,  are 
called  corresponding  parts. 

Hence,  it  may  be  inferred  that  any  part  (angle  or  side)  of 
one  of  two  congruent  triangles  is  equal  to  the  corresponding 
part  of  the  other. 

In  problem  2,  triangle  A'B'C  was  constructed  so  that 
one  side  and  the  two  adjacent  angles  were  equal  to  one  side 
and  the  two  adjacent  angles  of  triangle  ABC.  It  was  seen 
that  the  two  triangles  can  be  made  to  coincide  (are  congruent). 

4.  It  can  be  proved,  without  actually  placing  one  triangle 
over  the  other,  that  any  two  triangles  are  congruent  if  a  side 
and  the  two  adjacent  angles  of  one  triangle  are  equal  to  a 
side  and  the  two  adjacent  angles  of  the  other. 

The  proof  of  this  theorem  may  be  reasoned  out  thus:  Let 
ABC  and  A'B'C  (Fig.   252)  be  any  two  triangles  having 

AB=A'B' 
ZA  =  ZA' 
ZB  =  ZB'. 

Imagine  triangle  ABC  placed  on  A'B'C,  the  side  A  B  coin- 


B         A 

Fig.  252 


ciding  with  side  A'B'.     (Why  can  A  B  be  made  to  coincide 
with  A'B'  ?) 


332 


First-  Year  Mathematics 


Since  ZA=ZA',  the  side  AC  falls  along  A'C  and  the 
point  C  somewhere  on  A'C. 

Since  ZB=ZB',  the  side  B  C  falls  along  B'C  and  the 
point  C  somewhere  on  B'C 

Since  C  must  be  on  both  lines  A'C  and  B'C,  it  must  fall 
on  the  point  of  intersection  of  A'C  and  B'C.     (Why  ?) 

Therefore,  C  falls  on  C. 

Then  the  triangles  ABC  and  A'B'C  coincide  throughout 
and  are  congruent. 

In  the  same  way  in  which  ABC  and  A'B'C  were  made 
to  coincide,  any  two  triangles  can  be  made  to  coincide  if  a 
side  and  two  adjacent  angles  of  one  triangle  are  equal  to  a 
side  and  the  two  adjacent  angles  of  the  other. 

290.  Theorem:  If  in  two  triangles  a  side  and  tJie  two 
adjacent  angles  of  tfie  one  are  equal  respectively  to  a  side  and  the 
two  adjacent  angles  of  the  other,  the  triangles  are  congruent. 

I.  Prove:  In  two  right  triangles  if  the  hypotenuse  and 
a  side  of  the  one  are  equal  respectively  to  the  hypotenuse  and  a 
side  of  the  other,  the  triangles  are  congruent. 

Let  the  hypotenuse,  c,  and  the  side,  a,  of  one  triangle 
(Fig.  253)  be  equal  respectively  to  the  hypotenuse,  c,  and  the 
side,  a,  of  the  other  triangle. 


Fig.  253 

To  prove  the  triangles  congruent. 

Proof.     Place  the  triangle  at  the  left  so  that  the  equal  sides, 
a,  coincide,  as  shown. 


Polygons,  Congruent  Triangles,  Radicals 


333 


The  line  6'+ &  is  straight.     Why? 

The  figure  formed  of  the  two  triangles  is  an  isosceles  tri- 
angle.    Why  ? 

Complete  the  proof. 

2.  Give  the  proof,  using  Fig.  254. 


Fig.  254 

3.  In  triangles  RST  and  XYZ  (Fig.  255),  the  parts 
marked  in  the  same  way  are  equal.  Show,  as  in  problem  4, 
p.  331,  that  triangles  RST  and  XYZ  are  congruent. 


How  do  the  sides  T  R  and  X  Z  compare  in  length  ?     T  S 
and  Z  Y  ? 

How  do  angles  T  and  Z  compare  in  size  ? 

4.  In  triangle  D  E  F  (Fig.  256),  EG  bisects  angle  E  and 
makes  equal  angles  with  D  F  (i.  e,  is 
perpendicular  to  D  F).  What  parts 
of  the  two  triangles  are  known  to  be 
equal  ?  Show  that  triangles  D  E  G 
and  E  F  G  are  equal  in  all  respects. 
Hence,  side  DE 
D=  angle  F. 


c 

Fig.  256 
side  E  F;  side  D  G  =  side  G  F;  and  angle 


334 


First-Year  Mathematics 


5-  In  triangles  ABC  and  A'B'C  (Fig.  257)  the  parts 
marked  in  the  same  way  are  equal.  Prove  that'C  A  =  C'A'. 
What  other  parts  are  equal?     Give  reason. 


U  A 


X.. 


Fig.  257 

6.  In  Fig.  258,  A  B  is  equal  to  B  C  and  the  angles  at  A 
and  C  are  equal  as  indicated.  What  other  angles  of  the  tri- 
angles A  E  B  and  BCD  are  equal  ?  Prove  the  triangles  con- 
gruent, and  angle  A  E  B  equal  to  angle  B  D  C. 


/d-^° 


i^^i 


Fig.  258 


Fig.  259 


7.  In  Fig.  259,  a=b  and  c=d.     Prove,  by  means  of  con- 
gruent triangles,  that  R  S=U  T,  and  R  U  =  S  T. 

8.  In  the  parallelogram  A  B  C  D,  Fig.  260,  x=x',  y=y', 
AB  =  DC.    Prove  that  A  0  =  0  C,  B  0  =  0  D. 


Fig.  260 


9.  Triangles  R  S  T  and  R'S'T'  (Fig.  261)  are  congruent. 
Find  X  and  the  corresponding  sides  R  T  and  R'T'. 


Polygons,  Congrueni  Triangles,  Radicals 


335 


lo.  The  triangles  in  Fig.  262  are  congruent.     Find  s  and  /. 
How  long  is  B  C  ?    A  B  ? 


a  A      2t*i9 

Fig.  262 


II.  Draw  a  triangle,  as  ABC  (Fig.  263).  Construct  a 
triangle  having  the  sides  equal  respectively  to  the  sides  of 
triangle  ABC. 


...0^ 


Fig.  263 

On  a  line,  as  D  E,  lay  off  A'B'  equal  in  length  to  A  B. 

With  A'  as  center  and  a  radius  equal  to  A  C,  draw  an  arc  F  G 
above  D  E. 

With  B'  as  center  and  radius  equal  to  B  C,  draw  an  arc  meeting 
arc  F  G  at  C. 

Draw  lines  A'C  and  B'C. 

Triangle  A'B'C  is  the  required  triangle. 

12.  Prove  that  triangle  A'B'C,  Fig.  263,  is  congruent  to 
triangle  ABC. 

Proof.  In  triangle  ABC  (Fig.  263),  with  A  as  center  and 
A  C  as  radius,  draw  a  semicircle  x  (see  Fig.  264). 


In  A  A  B  C  (Fig.  263),  with  B  as  center  and  B  C  as  radius, 
draw  a  semicircle  y  (see  Fig.  264). 


33^ 


First-  Year  Matliemaiics 


In  ^A'B'C  (Fig.  263),  extend  arcs  through  C,  com- 
pleting semicircles  x'  and  y  (see  Fig.  264). 

Imagine  triangle  ABC  placed  on  A'B'C.  Then  A  B  can 
be  made  to  coincide  with  A'B'.     (Why  ?) 

Since  A  C  is  equal  to  A'C  (why?),  semicircle  x  coincides 
with  semicircle  x',  for  circles  drawn  with  the  same  center  and 
radius  coincide. 

Therefore  point  C  must  fall  somewhere  on  the  semicircle 
x\     (Why  ?) 

Since  B  C  is  equal  to  B'C,  semicircle  y  coincides  with 
semicircle  y',  and  point  C  lies  somewhere  on  semicircle  y. 

Therefore  point  C  must  be  on  the  point  of  intersection  of 
x'  and  y. 

Then  A  C  coincides  with  A'C,  and  B  C  with  B'C. 

The  triangles  coincide  throughout  and  are  congruent. 

13.  Two  triangles  as  in  Fig.  265  or  Fig.  266  have  three 
sides  of  the  one  equal  to  the  corresponding  sides  of  the  other. 


Prove  as  in  problem  12  that  A  A  B  C  and  l\J)  E  F  are  con- 
gruent. 

291.  Problems  11  and  12  estabUsh  the  truth  of  the  follow- 
ing theorem: 

If  two  triangles  have  three  sides  of  one  equal  to  the  three 
corresponding  sides  of  the  other,  the  triangles  are  congruent. 


Polygons,  Congruent  Triangles,  Radicals  337 

292.  We  have  thus  far  proved  three  theorems  on  con- 
gruent triangles,  viz. : 

I.  If  two  sides  and  the  included  angle  of  one  triangle  are 
equal,  each  to  each,  to  two  sides  and  the  included  angle  of  another 
triangle,  the  triangles  are  congruent.    See  §145,  p.  202. 

II.  //  tivo  angles  and  the  included  side  of  one  triangle  are 
eqical,  respectively,  to  two  angles  and  the  included  side  of  another, 
the  triangles  are  congruent.     See  §290,  p.  332. 

III.  If  three  sides  of  one  triangle  are  equal,  respectively,  to 
the  three  sides  of  another,  the  triangles  are  congruent.  See 
§291,  p.  336. 

These  three  theorems  on  congruent  triangles  are  used  in 
proving  other  theorems  in  this  chapter.  The  statements  of 
the  theorems  should  therefore  be  carefully  memorized  by  the 
student. 

293.  Circles  and  Triangles 

I .  Prove  the  theorem :  If  a  point  is  on  the  perpendicular 
bisector  of  a  line,  it  is  equally  distant  from  the  extremities  of 
the  line.* 

Let  L  M  be  perpendicular  to  A  B  at  the  middle  point  O, 
and  P  be  any  point  on  L  M. 

To  prove  P  equally  distant  from 
A  and  B. 

Proof.    Draw  P  A  and  P  B  and    ^ 
prove  triangles  P  O  A  and  P  O  B 
congruent.     (§I45-) 

ThenPA  =  PB.     Why? 


B  A' 


}t 


zr — 


ff 


But  P  is  any  point  on  A  B.  ^^^  267 

Therefore,  any  point  on  the  per- 
pendicular  bisector   of   a    line    is   equally  distant    from   the 
extremities  of  the  line. 

*  In  the  statement  of  this  theorem,  and  of  some  others  following,  it 
is  implied  that  there  is  only  one  perpendicular  bisector  of  a  line. 


338 


First-Year  Mathematics 


Fig.  268 


Prove  the  theorem  in  i,  using  Fig.  268. 

Prove  the  theorem:  If  a  point  is  equally  distant  from 
the  extremities  of  a  line,  it  is  on  the  perpen- 
dicular bisector  of  the  line. 

Let  P  be  any  point  equally  distant  from 
the  extremities  of  the  line  A  B ;  that  is, 
PA=PB.     (Fig.  269.) 

To  prove  that   P  is  on  the  perpendicular 
bisector  of  A  B. 
Proof.   From  O,  the  mid- 
point  of  A  B,   draw    a    Une 
to  P. 

Prove  triangles  P  O  A  and 
P  O  B  congruent.     (§291.) 

Then  x=y    and    P  O   is 
perpendicular  to  A  B.     Why  ? 
But  P  O  also  bisects  A  B. 
Why? 

Therefore  P  is  on  the  perpendicular  bisector  of  A  B. 
But  P  is  any  point  equally  distant  from  the  extremities 
of  AB. 

Therefore,  any  point  that  is  equally  distant  from  the  ex- 
tremities of  a  line  is  on  the  perpendicular  bisector  of  the  line. 

4.  Prove  the  theorem  in  3,  using  Fig.  270;  Fig.  271. 


Fig.  269 


— s 


p 

Fig.  270 


Fig.  271 


5.  Assuming  that  the  perpendicular  bisectors  X  M  and  Y  R 
of  two  sides  of  triangle  ABC  meet  at  O,  Fig.  272,  prove 

(i)  OB=OC,  OC  =  OA,  andOB  =  OA. 


Polygons,  Congrtient  Triangles,  Radicals 


339 


(2)  O  is  on  the  perpendicular  bisector  of  A  B. 
Draw  a  circle  with  O  as  center,  and  O  B  as  radius,  and  notice 
that  the  circle  passes  through  C  and  A. 


Fig.  272 

A  circle  passing  through  the  three  vertices  of  a  triangle  is 
circumscribed  about  the  triangle. 

6.  Draw  a  triangle  having  an  obtuse  angle.  Circumscribe 
a  circle  about  the  triangle. 

Construct  the  perpendicular  bisectors  of  two  sides  (see  problem  15, 
p.  147).  Prove  that  the  paint  where  the  perpendicular  bisectors  meet 
is  equally  distant  from  the  three  vertices. 

Notice  the  position  of  the  center  of  the  circle  with  respect 
to  the  triangle. 

7.  Circumscribe  a  circle  about  a  right  triangle.  Notice 
the  position  of  the  center  of  the  circle  with  respect  to  the 
triangle. 

In  problem  5,  where  does  the  center  of  the  circle  lie  with 
respect  to  the  triangle  ? 

294.  The  distance  from  a  point  to  a  line  is  the  length  of 
the  perpendicular  from  the  point  to  the  line. 

The  symbol  _L  stands  for  "perpendicular,"  "perpendicular 
to,"  or  "is  perpendicular  to." 

The  symbol  AI-  stands  for  "perpendiculars." 

I.  Prove  the  theorem: 


340 


First-  Year  Malhematics 


If  a  point  is  on  tJie  bisector  of  an  angle  it  is  equidistant  from 
the  sides  of  the  angle. 

Let    M  O    be    the    bisector   of 
A  O  B  and  P  be  any  point  on  M  O. 
To    prove   P   equidistant    from 
A  O  and  B  O. 

Proof.     Draw  P  C    _L  A  O  and 
PD  X  BO. 

Prove     triangles    P  O  C     and 
POD  congruent.     (§292.) 
Why? 

From 


Fig.  273 

ThenPC  =  PD. 

2.  Construct  the  bisector  of  an  obtuse  angle  A  O  B. 


\ 


Fig.  274 


any  point  P  on  the  bisector,  construct  perpendiculars,  P  C  and 
P  D,  to  the  sides.     Draw  a  circle 
with  P  as  center,  that  passes  through 
C  and  D. 

The  sides  A  O  and  B  O  (Fig.  274) 
are  tangent  to  the  circle. 

3.  Prove  the  theorem: 
If  a  point  is  equidistant  from  tlie 

sides  of  an  angle  it  is  on  the  bisector 
of  the  angle. 

Let  P  be  any  point  equidistant  from  the  sides  of  angle 
A  O  B;  that  is,  P  C  X  O  A,  and  P  D  ^ 
OB,  andPC=PD. 

To  prove  that  P  is  on  the  bisector  of 

angle  A  O  B. 

Proof.  Draw  a  straight  line  through 
P  and  O.  Prove  triangles  P  O  C  and 
POD  congruent  (problem  i,  p.  332). 

Then  x=y  and   P  O  bisects  A  O  B. 
Why  ? 
Therefore  P  is  on  the  bisector  of  A  O  B. 

4.  The  bisectors  C  M  and  A  L  of  two  angles  of  triangle 


Polygons,  Congruent  Triangles,  Radicals 


341 


ABC  meet  at  O.     Perpendiculars  are  drawn  from  O  to  the 
sides. 


Prove    (i)  O  X  =  0  Y,  O  Y  =  0  Z,  and  O  Z  =  0  X. 
(2)  O  is  on  the  bisector  of  angle  B. 

Draw  a  circle  with  O  as  center,  and  O  X  as  radius,  and 
notice  that  the  circle  passes  through  Y  and  Z. 

Polygons  is  a  general  name  for  triangles,  quadrilaterals, 
pentagons,  hexagons,  etc.  Polygon  is  another  name  for  an 
n-gon. 

A  circle  entirely  within  a  polygon,  all  sides  of  which  are 
tangent  to  the  circle,  is  inscribed  in  the  polygon. 

5.  Draw  a  triangle  having  an  obtuse  angle.  Draw  the 
inscribed  circle.  Construct  the  bisectors  of  two  angles  (see 
problem  20,  p.  148).  Prove  that  the  point  where  the  bisectors 
meet  is  equidistant  from  the  three  sides  of  the  triangle. 

6.  Inscribe    a  circle  in  a  A 
right  triangle. 

7.  In  Fig.  277,  show  that 

Area  of  B  O  C  =  ^r  •  a 

Areaof  AOC=ir  •  h 

Area  of  A  O  B  =  Jr  •  c. 
Therefore,   area   of   A  B  C  = 
\ra-\-\rh-\-\rc 


Fig.  277 


342  First-Year  Mathematics 

8.  Letting  A  stand  for  the  area  of  triangle  ABC,  Fig.  277, 
and  s  stand  for  ^(a  +  6+c),  show  that 

A=s  '  r 

and  r=— . 
s 

Theorem:  The  radius  of  a  circle  inscribed  in  a  triangle  is 
equal  to  the  area  of  the  triangle  divided  by  one-half  of  the  per- 
imeter. 

Application  of  the  Theorems  on  Congruent  Triangles 

295.  The  theorems  on  congruent  triangles  §292  may  be 
used  in  proving  the  correctness  of  the  fundamental  construc- 
tions in  chap,  vii,  §111. 

^Jf'  I.  Prove  that  CF  (Fig.  174, 

/'iX  p.  145)  is  perpendicular  to  A  B. 

/'    '     \  Proof.     Draw  the  lines  D  F 

^    ~"  "^       ^       ^     ^    and  E  F  as  in  Fig  278. 

^^^-  278  Then  in  triangles  D  C  F  and 

E  C  F  the  following  relations  are  known. 

DC  =  CE  Why? 
EF=DF  Why? 
CF  =  CF. 

Therefore  triangles  D  C  F  and  E  D  F  are  congruent.  Why  ? 
ZDCF=ZECF.     Why?     CF_lDE.     Why? 

Hence  the  line  C  F,  if  constructed  as  in  §iii,  is  perpendic- 
ular to  A  B  at  the  point  C. 

The  symbol  .'.  means  "therefore,"  or  "hence." 

2.  Prove  that  the  construction  in  problem  8  (p.  146)  makes 
C  F  perpendicular  to  A  B. 

Proof.     In  Fig.  176  draw  D  C  and  C  E  as  in  Fig.  279. 
C  is  equally  distant  from  D  and  E.     Why  ? 


Polygons,  Congruent  Triangles,  Radicals  343 

.".   C   lies   on   the   perpendicular  bisector  of    D  E     (see 
problem  3,  p.  340). 

Since   F   is   equally  distant   from  C 

D  and  E  (why?),  F  lies  on  the  per-  /'\ 

pendicular  bisector  of  D  E.     (Why  ?)  /    \     \ 

.'.  the   perpendicular  bisector   of    ^ ^^ — }//   - V  .a 

D  E  contains  C  and  F.  ^;y. 

.•.  line  C  F  must  be  the  perpen-  yig.  279 

dicular  bisector  of  D  E,  for  through 
two  points  only  one  straight  line  can  be  drawn. 

.'.  C  F  is  perpendicular  to  A  B.     Why  ? 

3.  Give  proof  for  problem  15  (p.  147). 
In  Fig.   180,   draw  the  lines  A  F,  BE,  AG,   G  B   (see 
.^  Fig.  280). 

^/  '  ["""^s  Show  as  in  problem  2  that  F  G  is 

^"^         I  "^^        the  perpendicular  bisector  of  A  B. 

-^^X^        f^      ^^  Then  H  is  the  midpoint  of  A  B. 

N-i//''  Why? 

'    '  4.  Give  proof  for  problem  19  (p. 

147)- 
In  Fig.  181,  draw  lines  E  F  and  D  F.     Prove  that  triangles  B  E  F 
and  B  D  F  are  congruent. 

Then  angle  C  B  F  equals  angle  A  B  F.     Why  ? 

5.  Prove  problem  24  (p.  148). 
In  Fig.  185,  draw  G  H  and  O  M. 

6.  In  Fig.  281,  A  C  =  C  B  and  x=y. 

Prove  ZA=ZB. 

Prove  triangles  A  C  D 
and    BCD    congruent 
(§145)- 
Fig.  282  7.    In    triangle   ABC,     Fig.    282, 

AC=BC;    prove  ZA=ZB. 
Bisect  ZC  (see  §iii)  and  use  problem  6. 


344  First- Year  Mathematics 

Theorem  :  If  a  triangle  has  two  equal  sides,  the  angles  oj 
the  triangle  opposite  the  equal  sides  are  equal. 

8.  In  Fig.  283,  A  C  =  C  B.     C  D  is  drawn  from  the  vertex 
to  the  midpoint  of  A  B.     Prove  x=y 

z=u 
CDxAB. 
Theorem:   In  an  isosceles  triangle  the  line 
joining  the  vertex  to  the  midpoint  of  the  base 
bisects  the  vertex-angle,  and  is  perpendicular  to 
the  base. 

9.  If  two  angles  of  a  triangle  are  equal,  the  sides  opposite 
them  are  equal. 

In  this  statement,  it  is  assumed  that  two  angles  of  a  tri- 
angle are  equal.     From  this  it  is  inferred  that  the  sides  oppo- 
site them  are  equal.     It  is  to  be  proved  that  this  is  true. 
Let  A  B  C  be  a  triangle  having  ZA=  ZB. 
To  prove  AC =BC. 
Proof.     Bisect  ZC. 

ZA=ZB  Why? 
x=y  Why? 
r=s  Why? 

Prove  triangles  ADC  and  B  D  C  con-       ^(^ '-^ ^s 

gruent.  Fig.  284 

Whence  A  C  =  BC.     Why? 

10.  Prove  that  the  line  bisecting  the  angle  at  the  vertex 
of  an  isosceles  triangle  bisects  the  base,  and  is  perpendicular 
to  the  base. 

What  is  assumed  in  this  statement?  What  inference  is 
drawn  from  this  assumption  ? 

11.  Prove  that  the  altitude  of  an  isosceles  triangle  bisects 
the  base  and  also  the  vertex-angle. 

What  is  assimaed  here  ?  What  is  the  inference  from  the 
assumption  ? 


Polygons,  Congruent  Triangles,  Radicals  345 

12.  Prove:  A  diagonal  of  a  parallelogram  divides  it  into 
two  congruent  triangles. 

What  is  the  assumption  ?     The  inference  from  it  ? 
Given   a  parallelogram   A  B  C  D,   and   B  D   one   of    the 
diagonals. 

To  prove  that  triangles  A  B  D  and  C  B  D  are  congruent. 

Proof.     Since  A  B  C  D  is  a  parallelo-        d c 

gram  A  B  is  parallel  to  D  C.     (§285.) 
.*.  x=x'  (see  Summary,  p.  151) 
A  D  is  parallel  to  B  C.     Why  ? 
axid  y=y'.     Why?  Fig.  285 

.*.  Triangles  A  B  D  and  D  B  C  are  congruent.     Why? 

296.  In  every  theorem  one  or  more  assumptions  are  made. 
From  these  assumptions,  inferences  (conclusions)  are  drawn. 
The  assumptions  in  a  theorem  are  called  the  hypothesis  and  the 
inferences  the  conclusion. 

The  hypothesis  and  conclusion  together  contain  everything 
that  is  stated  in  the  theorem.  Nothing  can  be  in  the  hypothesis 
or  in  the  conclusion  unless  it  is  in  the  theorem. 

1.  State  the  hypothesis  and  conclusion  in  the  following 
theorems : 

(i)  page  332,  §290  (4)  page  336,  §291 

(2)  page  335,  problem  16  (5)  page  338,  problem  3 

(3)  page  332,  problem    5  (6)  page  340,  problem  3. 

2.  Prove:  The  opposite  sides  of  a  parallelogram  are  eqtuU. 

Observe  that  the  hypothesis  in  this  theorem  is:  A  figure  is  a  parallelo- 
gram; the  conclusion:   the  opposite  sides  of  this  figure  are  equal. 

The  hypothesis  and  conclusion  must  be  stated  always  with 
reference  to  the  particular  figure  used  in  the  demonstration. 
Hypothesis.     A  B  C  D  is  a  parallelogram.     (Fig.  286.) 
Conclusion.     A  B=D  C  and  A  D=B  C. 
Proof.     Draw  A  C. 


346  First-Year  Mathematics 

Then  triangle  A  B  C  is  congruent  to  A  D  C.     Why  ? 
.•.AB=DC 
andAD  =  BC.    Why? 

^t;-, r-^  The  symbol  ^  may  be  used  for 

"is  congruent   to"    or    "coincides 

with."       The     statement,    triangle 

Pic  286  ABC    is    congruent    to    triangle 

D  E  F,     may    be    written    thus: 

AA  B  C:^AD  E  F. 

Prove  the  following  theorems: 

3.  Parallel  lines  intercepted  between  parallel  lines  are  equal 
(Fig.  286). 

4.  The  diagonals  of  a  parallelogram  bisect  each  other. 
The  proof  for  this  theorem  is  suggested  in  problem  12,  p.  334. 

5.  A  parallelogram  having  two  adjacent  sides  equal  is  a 
rhombus. 

6.  The  diagonals  of  a  rhombus  are  perpendicular  to  each 
other. 

Hypothesis.  A  given  figure  is 
a  rhombus,  in  particular  A  B  C  D, 
and  A  C  and  B  D  are  the  diagonals 
of  A  B  C  D. 

Conclusion.  The  diagonals  of 
the  figure  are  perpendicular  to  each 
other,  i.  e.,  AC  is  perpendicular 
toBD. 

Proof,  DC  =  CB     Why? 

DO  =  OB     Why? 

Prove   ZDOC=ZBOC. 

Prove  OC-LDB. 

Then,  ACxDB. 


Polygons,  Congruent  Triangles,  Radicals 


347 


7.  The  diagonals  of  a  rectangle  are  equal. 
In  Fig.  288  prove  A  B  C^A  B  D  (§287). 

8.  If  two  adjacent  sides  and  the  in-      ' 
eluded  angle  of  one   parallelogram   are 
equal  respectively   to  two  adjacent  sides 
and  the  included  angle  of  another,  the     a 
parallelograms  are  congruent. 

In  Fig.  289,       AI^AF 

Aii^_\ir. 

Show  that  parallelogram  A  B  C  D  can  be  made  to  coin- 
cide with  parallelogram  A'B'C'D'. 


Fig.  288 


c 

>' « ^ 


Fig.  289 

Prove  the  following  theorems: 

9.  //  two  adjacent  sides  of  one  rectangle  are  equal  respec- 
tively to  two  adjacent  sides  of  another,  the  rectangles  are  con- 
gruent. 

10.  Two  squares  having  a  side  of  one  equal  to  a  side  of  the 
other  are  congruent. 

In  §§105  and  106  the  following  theorem  was  studied: 
Two  straight  lines  that  make  equal  corresponding  angles  with 
a  transversal  are  parallel. 

Thus  in  Fig.  160,  p.  140,  if  a=e,  then  A  B  ||  C  D.* 

11.  Prove  that  if  c=e  (Fig.  160),  then  a=e  and  A  B  ||  C  D. 

12.  Prove  that  if  c=e,  then  h=d  and  A  B  ||  C  D. 
*The  symbol  ||  means  "is  parallel  to,"  or  "are  parallel." 


348  Fir  si- Year  Mathematics 

13.  Prove  that  if  two  lines  make  equal  alternate  interior 
angles  with  a  transversal,  the  corresponding  angles  are  equal. 

14.  Prove  that  if  two  lines  are  cut  by  a  transversal  making 
the  sum  of  the  interior  angles  on  the  same  side  equal  to  two 
right  angles,  the  corresponding  angles  are  equal. 

In  Fig.   160  let  (i+e  =  i8o.     Then  d-\-a=i2>o.     There- 
fore a=e. 

15.  Prove  that  two  lines  making  equal  alternate  interior 
angles  with  a  transversal  are  parallel. 

16.  Prove  that  if  two  lines  are  cut  by  a  transversal  making 
the  sum  of  the  interior  angles  on  the  same  side  equal  to  two 
right  angles,  the  lines  are  parallel. 

17.  State  three  theorems  which  furnish  tests  for  parallel 
lines. 

iS.  If  two  sides  of  a  quadrilateral  are  equal  and  parallel, 
the  figure  is  a  parallelogram. 


Fig.  290 

Hypothesis.     B  C=A  D.     B  C  ||  A  D. 
Conclusion.     A  B  C  D  is  a  parallelogram. 
Proof.     Draw  A  C. 
Then  AA  B  C^A A  C  D.     Why  ? 
ZBAC=ZACD. 
A  B  II  CD. 
.*.  A  B  C  D  is  a  parallelogram.     Why  ? 

19.  A  rhombus  is  a  parallelogram.     (§§285  and  286.) 

20.  A  rectangle  is  a  parallelogram.     (§§285  and  287.) 

21.  A  square  is  a  parallelogram.     (§§285  and  288.) 


Polygons,  Congruent  Triangles,  Radicals 


349 


Fig.  291 


The  Equilateral  Triangle 

297.  The  altitude -lines  of  a  triangle  are  perpendiculars 

from  the  vertices  to  the  opposite  sides. 

1.  Prove  that  an  altitude-line  of  an 
equilateral  triangle  bisects  the  base. 

Show  by  §292  that  triangles  AOB  and 
A  O  C  (Fig.  291)  are  congruent. 

2.  Prove  that  a  line  drawn  from  the 
vertex  of  an  equilateral  triangle  to  the  mid- 
point of  the  opposite  side,  is  perpendicular  to  that  side. 

Use  §33,  page  41. 

The  medians  of  an  equilateral  triangle  are  also  altitude- 
lines. 

3.  Prove  that  the  median  to  one  side  of 
an  equilateral  triangle  bisects  the  angle 
opposite  that  side. 

4.  Find  the  altitude  and  the  area  of  the 
equilateral  triangle  (Fig.  292). 

5.  Find  the  altitude  and  the  area  of  an 
equilateral  triangle  whose  side  is  6;  8;  12;  3;  5. 

Extracting  Square  Roots 

298.  The  process  of  extracting  square  roots  of  numbers  is 
used  in  finding  approximate  values  of  irrational  numbers  such 
as  1/75,  1/27,  1/^,  V'y^,  met  with  in  the  preceding  problems 
4  and  5. 

1.  Give  the  squares  of  the  following  numbers: 

I,  2,  3,  4,  5,  6,  7.  8,  9 

10,  20,  30,  40,  50,  60,  70,  80,  90 

100,  200,  300,  400,  500,  600,  700,  800,  900. 

2.  Show  that  the  square 

(i)  of  any  one-digit  number  has  one  or  two  digits 


350  First-Year  Mathematics 

(2)  of  any  two-digit  number  has  three  or  four  digits. 

(3)  of  any  three-digit  number  has  five  or  six  digits. 

3.  Illustrate  by  squaring  456  that  if  a  number,  like  207,936, 
be  marked  oflf  into  two-digit  groups,  thus  2o'79'36',  the  num- 
ber of  groups  is  the  same  as  the  number  of  digits  in  the  square 
root. 

299.  The  numerals  of  arithmetic  express  numbers  as  so 
many  tens  and  so  many  units,  or  so  many  hundreds,  so  many 
tens,  and  so  many  units,  and  so  on. 

To  find  the  square  root  of  a  number  means  to  find  first  the 
tens,  then  the  units;  or  first  the  hundreds,  then  the  tens,  then 
the  units,  according  as  the  root  is  a  two-digit,  or  a  three-digit 
number. 

By  problem  3  (§298)  one  can  tell  at  the  outset  how  many 
digits  there  are  in  the  square  root  of  a  given  number. 

I.  How  many  digits  are  there  in  the  square  roots  of  the 
following  numbers  ? 

1,681,  2,116,  961,  81,  121,104,  19,881,  964,324. 

300.  To  understand  the  way  to  find  square  roots,  it  is 
necessary  only  to  see  how  the  tens  and  the  units  enter  into  the 
squares  of  numbers,  and  then,  to  learn  how  to  reverse  the 
order  of  steps. 

*  Arithmetically  Algebraically 

57  =  50  +  7  n=t  +  u 

572  =  (5o  +  7)2  n^  =  {t  +  u)^ 

=  502  +  2.50.74-72  =t^  +  2tU  +  U' 

=  502  +  (2  .   50  +  7)7  =t^  +  {2t  +  u)u 

=  2,500  +  700  +  49=3,249 

Reversing  the  Steps 

3.249  I  50  +  7  =  57.  root 


2,500=  c;o2 


trial  divisor,  2X50=100 

second  root  figure,         7  =     7 


50  +  7    =107 


749  =  remainder 

749  =  (2.  50  +  7)7 


Polygons,  Congruent  Triangles,  Radicals 


351 


Trial  divisor, 
Second  root  figure, 


2X<  =  2< 

u=u 


2t  +  U 


t'  +  2tu  +  u'  \t  +  u,  root, 
t'       =  sq.  of  tens 


2tu  +  u^  =  remainder 
2tu+u^  =  {2t+u)u 


I.  Extract  the  square  root  of  183,184,  omitting  all  unneces- 
sary figures. 

i8'3i'84'|  428,  square  root 


16 


80 

2 

82 

840 


231 
164 


6,784 


848     6,784 


Check:  428*  =  183,184. 

2.  Extract  the  square  roots  of  the  following: 

(i)      784  (5)     42,436  (9)  120,409 


(2)  1,521 

(3)  2,209 

(4)  2,401 


(6)  52,441 

(7)  99,225 

(8)  112,896 


(10)  281,961 

(11)  389,376 

(12)  515,524. 


301.  Proceed  with  decimals  precisely  as  with  whole  num- 
bers, carrying  the  two-digit  grouping  toward  the  right  of  the 
decimal  point,  and  bringing  down  always  the  next  two  figures 
of  the  given  number. 

I.  Extract  the  square  roots  of  the  following: 

19.36  1-5876         0361 

2.89  2.1316  51.2656 

5.29  9.6721        .5329 

114.49  11.0889  78.4996 

210.64  -1225        .984064. 


352 


First-  Year  Mathemutics 


2.  Find  approximate  values,  to  four  decimal  places,  of  the 
following: 

V^2;  1/3;  l/i;  v/6;  1/7. 
Annex  zeros  and  proceed  as  with  decimals. 

302.  The  square  root  of  algebraic  expressions  is  extracted 
in  the  same  way  as  the  square  root  of  arithmetical  numbers. 

I.  Extract  the  square  root  of 

gx^—^oxz  —  i2Xy-\-4y^-\-2^z^-\-2oyz. 

For  convenience  arrange  the  polynomial  in  the  order  of  powers  of 
X,  y,  and  2  thus: 

square  root 
gx^  —  i2xy  +  4y^  —  ;^oxz+2oyz-\-2^z^  \  ^x—2y—^z 
gx' 


ist  trial 

divisor 
2d  root 


2  X  3-'c  =  6x 

.  root       ^  ~  1 2x^1  _ 
number  (      6:x;         ~  ^^ 


I  St  complete  ) 
divisor        ) 


6x—  2y 


—  i2xy+4y' 


—  1 2xy  +  4y'  = 


2y  •  {bx—  2y) 


2d  trial  divisor  2{;^x—2y)—6x  —  4y 
3d  root       (  -  3°-'^g  =      -  52 

number  (      6x 

2d  complete  divisor 


6x— 4V— 52 


o— 3o:x;s  +  2oyz  +  253* 


—  30A-3  +  2oy2  +  252*  =  —  52  •■  (6x —4y—  52 


Check:  (3.r—  2y—  ^z)^  =gx'  —  i2xy  +  4y^  —  30x2+  2o>'2+  252^. 

2.  Extract  the  square  roots  of  the  following: 
(i)   2^r^  —  'jors+4gs^ 

(2)  3(;4  +  i3:x;^  — i2;x;— 6x3+4 

(3)  x^+4xy+6xz+4y^  +  i2yz+gz' 

(4)  ga''-6ab''-6ac+b'i  +  2b''c+c^ 

(5)  X'*  +  2X  +  2X3  +  ^X^  +  I 

(6)  I +a^+a^  +  2a -2a3  — 2a4 

(7)  gz'  —  i2az+4a^+c'+6cz—4ac 


Polygons,  Congruent  Triangles,  Radicals  353 

(8)  r^'  +  zs+r'*  — 2r3  +  ior2  — lor 

(9)  s^  —  2r^s^-\-2s^  —  2r^-\-r^-\-i 

(10)  dtX^°—i\x'^y-{-/^^z^+y^+z^  —  2yz3. 

303.  Frequently  the  work  of  approximating  the  roots  of 
numbers  that  are  not  squares  is  much  shortened  by  simplifying 
the  irrational  numbers  before  extracting  the  roots. 

I.  In  an  equilateral  triangle  with  side  s  and  altitude  h 
show  that 

\i  s=  4,  then  h  =  1/12 
if  ^  =  10,  then  ^  =  1/75  ; 

if  5=  8,  then  h  =  V^ 
if  5  =  14,  then  A  =  ]/i47. 
Instead    of   approximating   directly   the   values   of    1/12^, 
1/75,  1/48,  V  147,  we  may  approximate  directly  only  1/3, 
and  multiply  the  result  by  2,  5,  4,  7,  respectively,  thus: 

1/12  =21/3  =  2X1.732  +  =  3.464  + 
V^75  =5^3  =  5X1.732  +  =  8.660+ 
1/48  =41/3=4X1.732  +  =  6.928  + 
1/147  =  71/3  =  7X1 .732 +  =12.124  +  . 

304.  In  reducing  the  irrational  numbers  |/i2,  7/75,  etc., 
in  §303,  the  following  principle  is  used: 

The  square  root  of  a  product  is  equal  to  the  product  of  the 
square  roots  of  the  factors. 

1.  Prove,  by  multiplying,  that  l/3*'5=3  •  V $. 

3  •  /S  •  3  •  v/5  =3  •  3  VS  •  1/5  =  3'  •  5-     Why  ? 

2.  Prove  the  following  by  multiplying: 

(i)  1/2^-  5^=2  •  s  (4)  y^r '  5=7  •  ^~s 

(2)  l/^^y^=:v>'  (5)   V\  -25  =  2-5 

(3)  l/a^6^=a6  (6)  1/4-7  =  21/7 


354  First-Year  Mathematics 

(7)  1/16 -3=4  •  \^Z  (10)  VTs  =  S'^Z_ 

(8)  1/12  =  2  .  1/3  (11)  1/108=61/3 

(9)  1/27=31/3  (12)  l/a'6'<;=a6l/c. 

3.  Reduce  1/18. 

V'i8  =  l/'9  •  2  =  1^9  •  1/2=3/2. 

4.  Reduce  y^2a^b^x3. 

y  ^2a3b5x3  =  V  i6a'b4x'  •  ^abx=v  i6a^b*x^  •  V^2a6a;  =  4a6*K  2a63C. 

5.  Reduce   the   following   irrational   numbers,    using   the 
principle  in  §304: 

(1)  1/45  (8)  Vl2Sx3y 

(2)  1/^  (9)  3i/75jc5^»° 

(3)  1/28  (10)  l/ya'-uab  +  yb' 

(4)  1/48  (11)  Vsx3{a-b)5 

(5)  1/125  (12)  1/270^^^186^ 


(6)  1/200  (13)  l/3^='(54 -45^/2 +4^4) 

(7)  1/288 


(14)  \'s-^- 

305.  It  should  be  kept  in  mind  that  the  square  root  of  a 
sum  is  not  equal  to  the  sum  of  the  square  roots  of  the  separate 
terms. 

1.  Prove,  by  multiplying,  that  V^a'  +b'  is  not  equal  to  a +  6. 
Show  that  {a  +  b)(a  +  b)  is  not  equal  to  a^  +  b'. 

2.  Prove  the  following,  by  multiplying: 

(i)  Vx'—y^  is  not  equal  to  x—y 

(2)  1/9  +  16  is  not  equal  to  3+4 

(3)  l/a'— 6  is  not  equal  to  a— l/ft. 

3.  Find  an  approximate  value  of  the  side  s,  and  the  area, 
of  the  equilateral  triangle  (Fig.  293). 


Polygons,  Congrtient  Triangles,  Radicals 
In  triangle  A  O  B 


355 


S' 


5^  = 

— +36.    Why? 

Show  that  s  = 
and  area  A  B  C  = 

=  6.928+ 
=  20.784  +  . 

A 

A 

A 

/  •  \ 

Fig.  293 

Fig.  294 

4.  Find  an  approximate  value  of  a  side,  and  the  area,  of 
an  equilateral  triangle  having  an  altitude  3;  9;  12;  15;  18. 

5.  Find  an  approximate  value  of  a  side,  and  the  area,  of 
the  equilateral  triangle  (Fig.  294). 

Show  that  5  =  / J^a 
l/I|tt  =  >/^  .  1/4  =  101/4.     Why? 
It  is  shown  in  the  following  problems  6  and  7,  that 
>/ J  =  1/1^1/3. 
Then  5  =  10(1 -M.732  +  )  =  s. 773  + 

and  area  A  B  C    =14.430  +  . 

6.  Prove  the  following  by  multiplication: 

(i)  (f)^=iV  (5)  m'=m 

(2)  {iy=u  (6)  (tV)'-tH 

(3)  (i)^=lf  (7)  (f)^=¥ 

(4)  {^y=TU  (8)  (f)^=¥- 

7.  Give  the  values  of  the  following  square  roots: 

(i)  ^^  (5)  y^  (9)  i^w 

(2)  l/il  (6)  1/^  (10)  V/|H 

(3)  i/f!  (7)  i/^  (")  i^Sf 

(4)  l/^^  (8)  i/S  (i2)l/fii. 


35^  First- Year  Mathematics 

306.  It  is  clear  from  6  and  7,  p.  355,  that  the  square  root  of 
a  common  fraction  is  the  square  root  of  the  numerator  divided 
by  the  square  root  of  the  denominator. 

I.  Give  the  following  square  roots. 


^)      " 


\b 


3) 


^¥. 


4)  \ 


c* 

a^x'^ 


\  (m  — 


{m—ny 


(7) 
(8) 
(9) 

u 

\ 

^^ 

r 

\ 

.1 

C2  +  2CJ+)/* 

>w^  +  2mw+«» 
ioo(a  +  6)* 

gy'+6yz-\-z' 

(Tr\\ 

iUic  +  3dy 

^lu; 

1 6  {4X'  —  1 2:x;y  +  gy') 

^TT^ 

i6a^64c2(w-n)^ 

yj^) 

8ir6(/2(r+5)6 

(^r,\ 

(6:x;-5>.)6(a  +  3<i)4 

5(a  +  6)^  \  (sa-cy{2r-ssy 


2.  Find  approximate  values,  to  four  decimal  places,  of  the 
following: 

307.  To  find  an  approximate  value  of  t/|  we  may  ap- 
proximate the  square  root  of  2,  obtaining  1.4142,  and  the 
square  root  of  3,  obtaining  i  .7320.  Divide  i  .4142  by  i  .7320, 
obtaining  .8165. 

But  the  following  method  of  finding  an  approximate  value  of 
t/|  is  shorter  and  gives  a  safer  approximation: 


V§  =  l/§-f  =  l/-|.     Why? 
l/l  =  T/r76=^l/6.      Why? 
•'•1^  =  ^X2. 4492  + =  .8164  +  . 


Polygons,  Congruent  Triangles,  Radicals  357 

By  the  first  method  we  obtain  the  square  roots  of  two 
numbers,  and  then  divide  the  root  of  the  numerator  by  that 
of  the  denominator. 

By  the  second  method,  we  reduce  the  radical  number  first, 
then  find  the  square  root  of  only  one  number,  and  divide  by  3 

1 .  Find  an  approximate  value  of  v\ . 

Vl=VY^=VY^2,  =  h^~2,-  ^112,  +  - 

\  2a^ 

2.  Find  an  approximate  value  of  aI  —  . 

11  (^X 


1 2a*  2a*     X         I  a4  o*        /-       /-     .4714a*   /- 

\  — =\ =  \/— T  •  2X=—  '  V 2  •  Vx- Vx, 

\  gx       \  gx     X      \  gx^  ^x  x 

approximately. 

3.  State  a  rule  for  simplifying  irrational  numbers  like  those 
in  problems  i  and  2. 

308.  To  simplify  an  irrational  number  which  is  the  square 
root  of  a  common  fraction  whose  denominator  is  not  a  square : 

(i)  Multiply  denominator  and  numerator  by  a  number 
that  will  make  the  denominator  of  the  resulting  fraction  a 
square. 

(2)  Factor  and  reduce  to  simplest  form  as  in  §307. 
Simplify  the  following,  and  find  approximate  values  to  four 
decimal  places: 

1.  l/|  6.  l/i  12.  l/Tf 

2.  l/|  7.  l/j^  lyc^yz^ 
7                     8.1/1                     '^'^    '^'^ 


+6 


9-   J/f  14.  J- 


5.  v\  "•  ^^  ''•  ^''■fe^ 


\ 


358 


First- Year  Mathematics 


1 6,  Find  the  side  and  area  of  an  equilateral  triangle  having 
an  altitude  7;   2;  8;   10;   22. 

17.  In  the  equilateral  triangle,  Fig.  295,  show  that 

^  =  §^1/3  (i) 

and  area  ABC =^h'  1/3.  (2) 

18.  Using  equations  (i)  and  (2),  problem 
17,  as  formulas  find  the  side  and  the  area  of 
an  equilateral  triangle  having  an,  altitude  6; 
9;   12;   10;   14. 


Fig.  295 


19.  In  the  equilateral  triangle.  Fig.  295, 
show  that 


h~-j/3 


2 

s'     .- 


and  area  A  B  C=— 1/3. 

4 


(3) 
(4) 


20.  By  means  of  formulas  (3)  and  (4),  find  the  altitude  and 
the  area  of  an  equilateral  triangle  having  a  side  6;   10;   9;   7. 

5  — 

21.  In  the  equation  h  =  -\/l, ,  substitute  1/5  for  h,  and 


show  that  5  = 


21/5 
1/3 


21/5 


To  find  an  approximate  value  of        -  ,  we  may  find  an 

approximate  square  root  of  5,  obtaining  2.2360,  and  of  3, 
obtaining  1.7320.  Multiply  2.2360  by  2,  and  divide  the 
result  by  i  .7320,  obtaining  2.5814. 

The  following  method  of  finding  an  approximate  value  of 

2v/K 

-   is  shorter,  and  gives  a  safer  approximation: 

l^3_  _  _ 

2l/S      2l/s      1/3      21/5  •  1/3       21/15       1     y /-!• 

-^=-^  •  i-^=-^—^ — *L^=-K — 2=,i-^/Ye,  Give  reasons. 
V3       Vl      VZ  3  3         '"^    ^ 


Polygons,  Congruent  Triangles,  Radicals  359 

1/75=3.8729  + 
•■•§^'15  =  2.5818  +  . 

22.  In  the  following,  multiply  numerator  and  denominator 
by  a  number  that  will  make  the  denominator  of  the  resulting 
fraction  a  rational  number.  Simplify  and  find  approximate 
values  to  four  decimal  places: 

(^);73         ^''V^         '''St 

(3)  4^      (6)  r0      (9)  '-^: 

^^    2v/i2         ^       i/io  ^^'    i/32a 

23.  Solve  the  equation 

for  5  in  terms  of  h,  and  find  the  value  of  5  for  h  =  \/io. 

24.  Solve  the  equation 

for  h  in  terms  of  s,  and  find  the  value  of  h  for  5=1/7. 

Problems  Involving  Radicals 

1.  Find  the  area  A  of  an  equilateral  triangle  in  terms  of 
the  side  5. 

2.  Using  the  equation  obtained  in  problem  i,  find  the  side 
of  an  equilateral  triangle  whose  area  is  25  sq.  in.;  161/3  sq-  in-; 
A  square  inches. 

3.  Find  the  area  A  of  an  equilateral  triangle  in  terms  of 
the  altitude  h. 


36o 


First-  Year  Mathematics 


4.  Using  the  equation  obtained  in  problem  3,  find  the 
altitude  of  an  equilateral  triangle  whose  area  is  171/3  sq.  ft.; 
35  sq.  ft.;  A  sq.  ft. 

5.  A  plot  of  ground  is  to  be  staked  oflf  in  the  form  of  an 
equilateral  triangle  covering  an  area  oi  100  sq.  ft.  How  long 
must  the  side  be  ?    The  altitude  ? 

6.  How  many  degrees  are  there  in  each  interior  angle  of 
an  equilateral  triangle  ? 

Show  that  six  equal  equilateral  triangles  may  be  so  placed  in  a  plane 
about  a  point  that  they  just  fill  the  angular  magnitude  about  the  point. 

How  many  degrees  are  there  in  each  interior  angle  A,  B,  C,  etc.,  of 
the  polygon  thus  formed  (Fig.  296)  ? 

Show  that  the  polygon  is  a  regular  hexagon. 


A B 

Fig.  296 

7.  Draw  a  regular  hexagon  starting  with  an  equilateral  tri- 
angle  having  a   side  3.     Circumscribe  a 

circle   about  the  hexagon.     How  long  is 
the  radius  of  the  circle  ? 

8.  Letting  A  be  the  area  of  a  regular 
hexagon,  and  R  the  radius  of  the  circum- 
scribed circle  (Fig.  297),  show  that  a  side 
of  the   regular  hexagon  is   R,   and   that 


/ 

//      \ 

/ 

\\ 

l\ 

A" 

/{ 

v\           / 
\\     / 

\ 

/I 

\\  / 

\ 

// 

\ 

Fig.  297 


9.  Find  the  area  of  a  regular  hexagon  whose  side  is  2 ;    i ; 

3;  4;  is;  ^- 


Polygons,  Congruent  Triangles,  Radicals 


361 


10.  Find  the   radius   of  a  circle  circumscribed   about   a 
regular  hexagon  whose  area  is  31/3;  6]/r5;  91/7;  A. 

1 1 .  Letting  r  be  the  radius  of  a  circle  inscribed  in  a  regula 
hexagon  (Fig.  298),  find  the  area  A  of  the  hexagon,  and  s,  th 

length  of  a  side  in  terms  of  r. 

12.  Find  the  value  of  A  and  r,  Fig. 
298,  in  terms  of  s. 

13.  Make  problems  that  can  b(. 
solved  by  the  aid  of  the  formulas  ob- 
tained in  problems  11  and  12. 

^^^'  ^98  j^    Show  that  a  side  5  of  a  square 

inscribed  in  a  circle  (Fig.  299)  is  equal  to  r|/2,  and  that  the 
area  A  is  equal  to  2r*. 

15.  Find  the  values  of  A  and  R  (Fig. 
299)  in  terms  of  S. 

16.  Make  problems  that  can  be  solved 
by  means  of  the  formulas  obtained  in 
problems  14  and  15. 

17.  Calculate  the  radius  of  the  circle 
circumscribed  about  a  square  whose  area  is  625;  200;  y^;  A. 

18.  The  diagonal  of  a  square  is  2  inches  longer  than  a 
side  X.     Find  the  area. 

19.  The  diagonal  of  a  square  is  b  inches  longer  than  a  side. 
Find  the  length  of  a  side. 

20.  Show  that  the  area  of  triangle  A  B  C  in  the  rhombus, 
Fig.  300,  is  equal  to  xy. 

See  problems  4  and  6,  p.  346,  and  problem  19,  p.  348. 

21.  Show  that  the  area  of  the  rhombus,  Fig.  3CX),  is  2xy. 
The    area    of  a    rhombus   is    one-half  the  product  of  the 

diagonals. 


Fig.  299 


362  First- Year  Mathematics 

22.  A  side  of  a  rhombus  is  15,  and  one  diagonal  is  24. 
Find  the  other  diagonal  and  the  area. 

23.  A  side  of  a  rhombus  is  12, 
and  one  diagonal  is  18.  Find  the 
area. 

24.  One  diagonal  of  a  rhombus 
is  twice    as    long  as  the  other,  and 

Fig.  ?oo  ^^^  ^^^^  is  ^28.     Find  the  length  of 

each  diagonal. 

25.  Find  the  area  of  an  isosceles  right  triangle,  the  hypote- 
nuse being  15;  3 1/7;  h. 


Algebraic  Problems  Based  on  Geometry 

I.  The  sides  and  angles  of  the  triangles  being  designated 
as  in  Fig.  301 ,  find  c,  x,  and  y. 


Fig.  301 
2.  In  the  two  triangles  of   Fig.    302,   a=a',  c=cf,  and 


ZB  =  ZB';    find  the  values  of  x,  y,  and  z  and  of  h,   ZA 
and  ZC: 


Polygons,  Congruent  Triangles,  Radicals 


363 


(i)  If    6=3^+17,    &'  =  73-2;y;     ZA=5x—i2°,    ZA'  = 
6o°-:!c;    ZC  =  i5o°-2Z,  and  ZC'  =  2(3z-5°). 

(2)  lib=y+l,y=^+g;   ZA=3{4X-^i7z),  ZA'=5f- 

2  5 

i5:x;;    ZC  =  23;x:— 32,  ZC'=8z  +  io3°. 

(3)  If  b=y(y-i),   y=s(y-7);     ZA=xix-2°),    ZA'  = 
5(2^-7°);    ZC=z(z-4°),  Ze  =  i6(z-6°). 

3.  With  angles  and  sides  of  two  triangles  of  values  that 
may  bq  designated  as  in  Fig.  303,  find  the  values  of  a,  c,  and  x. 


C     '"  ~3a*4c 

Fig.  303 

4.  In  Fig.  304  find  r  and  s. 

A 


Fig.  304 


(i)  if  6  =  28,  h'  =  2r-—,  a  =  2f,  and  a'=r  +  '^—  ; 

(2)  if  &=3/'  +  25  — I,  6'=68,  and  a  =  2r-\-2'^s,  a'=6o  ; 

(3)  if  h=6r—s—6,  6'  =  3(r+5)+2,  and  a=y-\-2s-\-2,  and 

25 

a'  =  5r -—I. 

3 


364 


First-  Year  MatJiematics 


5.  In  Fig.  305  the  triangles  have  sides  and  angles  of  values 
as  designated.     Find  x,  y,  and  z. 


Fig.  305 

6.  The  triangles  in   Fig.   306   have   sides  and  angles  of 
values  as  designated.     Find  x,  y,  and  z. 


Fig.  306 

7.  The  two  triangles  of  Fig.  307  are  congruent.  If  the 
values  of  the  angles  may  be  designated  as  shown,  what  are  the 
values  of  jc,  y,  and  z  ? 


Fig.  307 

8.  The  lengths  of  one  pair  of  opposite  sides  of  a  parallelo- 
gram are  %x-\-6y  and  12— y,  and  the  lengths  of  the  other  pair 

20C~\~V  OC  ~\~  2V 

of  opposite  sides  are and  i .     Find  x,  y,  and 

3  4 

the  lengths  of  the  sides  of  the  parallelogram. 

309.  The  corresponding  sides  of  triangles  whose  angles  are 
equal,  each  to  each,  are  proportional. 


Polygons,  Congrtient  Triangles,  Radicals  365 

For  example,  if  in  Fig.  308,  ZA=ZA',  ZB  =  ZB',  and 
ZC=ZC',  then  a:a'=b:b'=c:c'. 


The  ratio  of  any  pair  of  corresponding  sides,  as  a :  a',  b:  b', 
etc.,  is  called  the  ratio  of  similarity  of  the  figures. 

If  the  ratio  of  similarity  is  3,  find  the  values  of  x,  y,  and  z 
under  the  following  conditions: 

,    ,  V  3.  a=:ic+3'4-32  and  a'=4 

1.  a=^+z,  and  a'  =  io  +  ^  ^    ,  f    f^        ,  , , 

3  b  =  2X-\-y-\-oz  and  0'  =  7 

,  ,,  X  c =x -\- 2y -\- QZ  dsid  c'  =  q    ■ 

o=3:!C—z  and  o'=4  +  - 

3  4.  a=x+j4-z  and  a'  =  3 

c  =  7z-\-2X  and  c'  =  '4^±^  b  =  3x+4y-Sz  and  ^'  =  3 

3  c=4X+y  —  2Z  and  c'=4 

2.  a=x-\-y  3.nd  a' =  5— X  5.  a=3:c+;y+4Z  and  a'  =  7 
6  =  2ap+)'  and  b'  =  'j  —  2z  b=x+4y  +  2z  and  fe'  =  5 
c=3Z  +  2>'  and  c'=x+i  c=x+$y+;3z  and  c'=6. 


y 


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